#423094

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=efi,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,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, 330290);