#6541169

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1715 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)(7,16)(8,13)(9,14)(10,12)(11,15)(17,20)(18,19)(21,22,28,25)(23,33,30,36)(24,35,29,34)(26,31,27,32);(5,7,11,10)(6,12,15,16)(8,18,9,17)(13,20,14,19)(21,23,27,34,28,30,26,35)(22,29,31,36,25,24,32,33);(2,4)(5,8,11,9)(6,13,15,14)(7,17,10,18)(12,19,16,20)(21,24,26,36,28,29,27,33)(22,30,32,34,25,23,31,35);(5,9,11,8)(6,14,15,13)(7,17,10,18)(12,19,16,20)(21,25,28,22)(23,24,30,29)(26,31,27,32)(33,35,36,34);(5,10,11,7)(6,12,15,16)(8,18,9,17)(13,19,14,20)(21,26,28,27)(22,31,25,32)(23,34,30,35)(24,36,29,33)

Polycyclic presentation:

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

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, 6448365);