#53364

order	256
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

619 conjugacy classes of subgroups; only the 395 normal subgroups are shown.

normal subgroups only

This group has 619 conjugacy classes of subgroups — too many to draw. Only its 395 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(3,9)(7,12)(10,15)(14,16)(17,18,21,22)(19,23,25,29)(20,27,28,24)(26,32,31,30)(33,34)(35,39)(36,43)(37,40)(38,41)(42,46)(44,48)(45,47);(2,6)(4,11)(7,12)(10,15)(18,22)(19,25)(20,28)(30,32)(33,35,38,36)(34,39,41,43)(37,42,45,44)(40,46,47,48);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31)(33,36,38,35)(34,39,41,43)(37,44,45,42)(40,46,47,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32)(33,37,38,45)(34,40,41,47)(35,42,36,44)(39,46,43,48);(1,4)(2,8)(3,10)(5,11)(6,13)(7,14)(9,15)(12,16)(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)(33,34)(35,43)(36,39)(37,40)(38,41)(42,48)(44,46)(45,47)

Polycyclic presentation:

2^8|a^2=f,b^2=g,c^2=fgh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#53364⋊C₃
maximal subgroup of	#97312, #179427, #179428, …
maximal normal subgroup of	#97312, #179427, #179428, …
maximal quotient of	#606838, #607221, #607227, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 46352);