#6672162

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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
prime radical	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1788 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6579358);