#6489576

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469989
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35292	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2924 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)(7,11)(8,12)(9,13)(10,14)(15,18)(16,19)(17,20)(22,27)(24,25)(28,34)(29,33)(30,32)(35,36);(5,7,10,17)(6,11,14,20)(8,16,9,15)(12,19,13,18)(21,22)(23,28)(24,27)(25,33)(26,29)(30,35)(31,36)(32,34);(2,4)(5,8,10,9)(6,12,14,13)(7,15,17,16)(11,18,20,19)(21,23)(22,28)(24,32)(25,30)(26,31)(27,34)(29,36)(33,35);(5,9,10,8)(6,13,14,12)(7,16,17,15)(11,19,20,18)(21,24,26,25)(22,27,29,33)(23,30,31,32)(28,35,36,34);(6,14)(11,20)(12,13)(18,19)(23,31)(28,36)(30,32)(34,35)

Polycyclic presentation:

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

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, 6396772);