#6519747

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499792
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1968 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=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, 6426943);