#425761

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1792 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)(9,10)(11,14)(12,16)(13,15)(17,18)(19,27)(20,25)(21,23)(22,26)(24,29)(28,31)(30,32);(1,2,3,4)(9,11,13,12)(10,14,15,16)(18,23)(19,28)(20,22)(24,31)(27,32)(29,30);(9,12,13,11)(10,14,15,16)(17,19,21,29)(18,24,23,27)(20,28,22,30)(25,32,26,31);(2,4)(6,8)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,32,27,31)

Polycyclic presentation:

2^9|b^2=ef,c^2=f,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, 332957);