#374189

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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

1722 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=fh,d^2=i,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, 281385);