#415275

order	512
nilpotency class	3
isoclinism	isoclinic class of #393940
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 ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1240 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=f,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, 322471);