#468065

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

1350 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)(7,10)(8,9)(11,12)(21,22,25,30)(23,27,26,29)(24,32,34,35)(28,33,36,31);(6,9)(10,12)(13,14,16,18)(15,19,20,17)(21,23,25,26)(22,27,30,29)(24,31,34,33)(28,35,36,32);(2,4)(5,7)(6,10)(8,11)(9,12)(13,15,16,20)(14,17,18,19)(22,27)(23,26)(28,35)(29,30)(31,33)(32,36);(7,11)(10,12)(21,24,25,34)(22,28,30,36)(23,31,26,33)(27,35,29,32)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=i,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^c=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi

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, 375261);