#373322

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×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

1210 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)(11,15)(12,14)(13,16)(17,18)(19,25)(20,22)(21,29)(23,26)(24,27)(28,31)(30,32);(1,2,4,3)(5,7,8,6)(9,11,13,12)(10,14,16,15)(18,22)(19,26)(21,27)(23,31)(25,30)(28,32);(9,12,13,11)(10,15,16,14)(17,19,27,32,20,28,21,26)(18,23,29,31,22,30,24,25);(1,3,4,2)(10,16)(14,15)(18,22)(19,28)(24,29)(26,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fi,d^2=efg,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

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, 280518);