#444595

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
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 ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1146 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)(6,9)(10,11)(13,14,16,18)(15,17,19,20)(21,22)(23,31)(24,30)(25,28)(26,29)(27,32)(33,36)(34,35);(5,6,8,9)(7,11,12,10)(15,19)(17,20)(21,23)(22,27)(24,32)(25,33)(26,34)(28,35)(29,36)(30,31);(2,4)(13,15,16,19)(14,17,18,20)(22,28)(25,26)(27,35)(29,30)(31,36)(33,34);(5,7,8,12)(6,10,9,11)(22,29)(23,32)(25,26)(27,35)(28,30)(31,36)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=g,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 351791);