#414875

order	512
nilpotency class	3
isoclinism	isoclinic class of #393940
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1330 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=e,d^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=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, 322071);