#429900

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

1166 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)(9,10,13,15)(11,14,12,16)(17,18)(19,24)(20,25)(21,23)(22,26)(27,32)(28,30)(29,31);(5,6)(7,8)(9,11,13,12)(10,14,15,16)(18,23)(19,27)(20,22)(24,31)(28,29)(30,32);(10,15)(14,16)(17,19,21,28)(18,24,23,30)(20,27,22,29)(25,32,26,31);(2,4)(6,8)(9,12,13,11)(10,16,15,14)(17,20,21,22)(18,25,23,26)(19,27,28,29)(24,32,30,31)

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=i,d^2=fi,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, 337096);