#6569754

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2131 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^d=eh

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, 6476950);