#6473335

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470901
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2348 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(2,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,25)(16,29)(17,22)(18,34)(19,24)(20,27)(21,31)(23,35)(26,39)(28,38)(30,40)(32,36)(33,42)(37,44)(41,43);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,27,19,28,35,44)(14,20,23,25,24,37,34,38)(16,17,31,33,30,32,26,43)(21,36,29,42,39,22,40,41);(7,11)(9,12)(13,16,18,31,19,30,35,26)(14,21,23,29,24,39,34,40)(15,17,27,33,28,32,44,43)(20,36,25,42,37,22,38,41);(7,11)(9,12)(13,17)(14,22)(15,26)(16,27)(18,33)(19,32)(20,29)(21,38)(23,41)(24,36)(25,39)(28,31)(30,44)(34,42)(35,43)(37,40);(6,10)(9,12)(16,30)(17,32)(21,39)(22,36)(26,31)(29,40)(33,43)(41,42)

Polycyclic presentation:

2^9|b^2=f,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=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, 6380531);