#6520445

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500443
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	#25797	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2272 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)(15,25)(16,30)(17,22)(18,23)(19,24)(20,29)(21,32)(26,35)(27,37)(28,36)(31,38)(33,40)(34,39)(41,43)(42,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,17,29,19,27,18,28)(14,20,23,25,24,36,22,37)(16,31,33,43,34,26,32,42)(21,38,39,44,40,35,30,41);(6,9)(7,11)(13,16,18,32,19,34,17,33)(14,21,22,30,24,40,23,39)(15,26,28,43,27,31,29,42)(20,35,37,44,36,38,25,41);(5,7,8,11)(6,10,9,12)(15,27)(20,36)(25,37)(26,31)(28,29)(35,38)(41,44)(42,43);(6,9)(10,12)(13,17,19,18)(14,22,24,23)(15,28,27,29)(16,32,34,33)(20,25,36,37)(21,39,40,30)(26,42,31,43)(35,44,38,41)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,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, 6427641);