#94960

order	512
nilpotency class	3
isoclinism	isoclinic class of #7943
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#95
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

976 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,7,8,5)(4,6)(9,10,11,13,12,14,15,16)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,31)(30,32);(1,3)(2,5)(4,8)(6,7)(17,19)(18,22,24,30)(20,27)(21,26)(23,25,28,31)(29,32);(3,8)(5,7)(19,26)(22,30)(25,31)(27,32)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gh,e^2=i,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,e^b=eh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239975, #72239977, #72257795, …
maximal normal subgroup of	#72239975, #72239977, #72257795, …
maximal quotient of	#72240001

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2156);