#412919

order	512
nilpotency class	3
isoclinism	isoclinic class of #393940
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

1060 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fi,c^2=e,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,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, 320115);