#372528

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1498 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=ei,d^2=fgi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,g^c=gi

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, 279724);