#445608

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

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

1062 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 352804);