#96334

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

852 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)(4,6)(5,8)(9,10,13,15,14,11,12,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,28)(27,30)(31,32);(1,3)(2,5,6,7)(4,8)(9,11,14,10)(12,15,13,16)(17,19,20,25,21,26,27,31)(18,22,24,29,23,28,30,32);(3,8)(5,7)(18,23)(22,28)(24,30)(29,32)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=g,e^2=g,f^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=cgh,d^b=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3530);