#6807804

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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 ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2045 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6715000);