#444072

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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

1056 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)(5,6,9,8)(7,11,12,10)(13,14,16,18)(15,17,19,20)(21,22,24,27)(23,29,31,26)(25,28,32,34)(30,35,36,33);(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,17,18,20)(23,30)(25,32)(26,33)(28,34)(29,35)(31,36);(5,6,9,8)(7,10,12,11)(14,18)(17,20)(21,23,25,30,24,31,32,36)(22,26,28,33,27,29,34,35);(2,4)(5,8,9,6)(7,10,12,11)(23,31)(26,29)(30,36)(33,35)

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=eg,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,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, 351268);