#6492397

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470161
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1898 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:

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

Polycyclic presentation:

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

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, 6399593);