#444773

order	512
nilpotency class	3
isoclinism	isoclinic class of #417442
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

1052 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)(7,9)(8,10)(11,12)(13,14,16,18)(15,17,19,20)(21,22,25,29,26,24,27,30)(23,31,33,35,32,36,34,28);(5,7,8,11)(6,9,10,12)(15,19)(17,20)(21,23)(22,28)(24,35)(25,34)(26,32)(27,33)(29,36)(30,31);(6,10)(9,12)(13,15,16,19)(14,17,18,20)(23,32)(28,35)(31,36)(33,34);(2,4)(21,24,26,22)(23,31,32,36)(25,29,27,30)(28,34,35,33)

Polycyclic presentation:

2^9|a^2=egi,b^2=f,c^2=gi,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,h^b=hi,e^d=ei,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, 351969);