#6508741

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499792
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	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1988 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=i,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi

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, 6415937);