#6687836

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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×#496	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1752 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)(5,6,10,14)(7,15,17,11)(8,12,9,13)(16,20,18,19)(21,22)(23,25)(24,33)(26,29)(27,28)(30,31)(32,36)(34,35);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,22,26,29)(23,30,32,35)(24,28,27,33)(25,34,36,31);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,23,26,32)(22,25,29,36)(24,31,27,34)(28,35,33,30);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(21,24,26,27)(22,28,29,33)(23,31,32,34)(25,35,36,30);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(21,25,26,36)(22,23,29,32)(24,30,27,35)(28,34,33,31)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fg,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6595032);