#18465

order	256
nilpotency class	3
isoclinism	isoclinic class of #18446
automorphism	Aut(G) = C₂×#72286737 (order 4096) · Inn(G) = C₂×#120 · Out(G) = C₂×#120
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
prime radical	#538	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

435 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 111 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) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

34 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#347426, #347427, #347428, …
maximal normal subgroup of	#347426, #347427, #347428, …
maximal quotient of	#168936, #168952, #168991, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 11453);