#438613

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

1002 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=egi,c^2=efi,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 345809);