#464780

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1308 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371976);