#450688

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

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 ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

(6,10)(11,12)(13,14,17,19)(15,18,16,20)(21,22,26,28,24,29,25,30)(23,31,34,35,32,27,33,36);(1,2)(3,4)(5,6,9,10)(7,11,8,12)(13,15,17,16)(14,18,19,20)(21,23)(22,27)(24,32)(25,33)(26,34)(28,36)(29,31)(30,35);(2,4)(5,7,9,8)(6,11,10,12)(13,16,17,15)(14,18,19,20);(5,8,9,7)(6,11,10,12)(22,28)(23,32)(25,26)(27,35)(29,30)(31,36)

Polycyclic presentation:

2^9|a^2=fhi,b^2=efi,c^2=efi,d^2=ei,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,h^d=hi

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, 357884);