#424534

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1236 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=ef,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,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, 331730);