#19093

order	256
nilpotency class	3
isoclinism	isoclinic class of #19086
automorphism	Aut(G) = C₂×#72286730 (order 4096) · Inn(G) = C₂×#120 · Out(G) = C₂×#120
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

391 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,12)(3,13,15,25)(4,17,19,26)(5,11,6,10)(8,20,23,16)(9,21,24,18)(14,22,28,32)(27,29,31,30)(33,34,38,41)(35,43,45,57)(36,40,48,55)(37,49,52,42)(39,50,54,46)(44,60,61,59)(47,58,51,56)(53,63,64,62);(1,3)(2,8)(4,14)(5,20)(6,16)(7,15)(9,22)(10,25)(11,13)(12,23)(17,31)(18,30)(19,28)(21,29)(24,32)(26,27)(33,35,38,45)(34,39,41,54)(36,44,48,61)(37,50,52,46)(40,53,55,64)(42,57,49,43)(47,63,51,62)(56,59,58,60);(1,4,7,19)(2,9,12,24)(3,14,15,28)(5,18,6,21)(8,22,23,32)(10,17,11,26)(13,27,25,31)(16,29,20,30)(33,36,38,48)(34,40,41,55)(35,44,45,61)(37,51,52,47)(39,53,54,64)(42,58,49,56)(43,59,57,60)(46,63,50,62);(3,15)(8,23)(13,25)(14,28)(16,20)(22,32)(27,31)(29,30)(34,41)(35,45)(40,55)(42,49)(44,61)(46,50)(56,58)(62,63)

Polycyclic presentation:

2^8|a^2=g,b^2=h,c^2=g,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fgh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fgh

Conjugacy classes & character table

31 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#250490, #250491, #250498, …
maximal normal subgroup of	#250490, #250491, #250498, …
maximal quotient of	#168924, #168928, #168930, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 12081);