#431098

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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1034 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,c^2=f,d^2=fi,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,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, 338294);