#96349

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₄³	C₂³
socle	C₂³	C₂×#95
prime radical	C₂⁴	#95
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gi,d^2=g,e^2=h,f^2=i,b^a=bd,d^a=dg,f^a=fgh,c^b=ch,d^b=dh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3545);