#8141

order	256
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#376
prime radical	C₂³	#117
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

147 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^8|a^2=e,b^2=f,c^2=d,d^2=g,e^2=h,b^a=bd,f^a=fg,c^b=ch,e^b=eg

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#128235, #128236, #135640, …
maximal normal subgroup of	#128235, #128236, #135640, …
maximal quotient of	#96660, #96661, #96663, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 1129);