#6529963

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500535
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1733 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)(21,22)(23,30)(24,29)(25,34)(26,28)(27,32)(31,35)(33,36);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,25,32,26,33,24,31)(22,27,29,30,28,35,34,36);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(22,28)(27,35)(29,34)(30,36);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,22)(23,27)(24,34)(25,29)(26,28)(30,32)(31,36)(33,35);(6,12)(10,13)(11,20)(18,19)(21,24,26,25)(22,29,28,34)(23,31,33,32)(27,30,35,36)

Polycyclic presentation:

2^9|b^2=f,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei

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, 6437159);