#426715

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
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

1194 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)(10,13)(14,16)(17,18)(19,24)(20,27)(21,23)(22,29)(25,26)(28,32)(30,31);(1,2)(3,4)(5,6)(7,8)(9,10,12,13)(11,14,15,16)(18,23)(24,29)(25,31)(27,32);(2,4)(6,8)(9,11,12,15)(10,14,13,16)(17,19,21,22)(18,24,23,29)(20,26,28,30)(25,32,31,27);(6,8)(10,13)(14,16)(17,20,21,28)(18,25,23,31)(19,26,22,30)(24,32,29,27)

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=ef,d^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,h^d=hi

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, 333911);