#6520292

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	#25733	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ 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)(13,14,19,23)(15,24,26,36)(16,21,30,40)(17,22,18,33)(20,31,35,27)(25,41,28,44)(29,38,32,39)(34,42,37,43);(5,6)(7,10)(8,9)(11,12)(13,15,18,27,19,26,17,31)(14,20,22,24,23,35,33,36)(16,28,29,42,30,25,32,43)(21,37,39,44,40,34,38,41);(1,2,3,4)(6,9)(7,11)(13,16)(14,21)(15,25)(17,32)(18,29)(19,30)(20,34)(22,39)(23,40)(24,41)(26,28)(27,43)(31,42)(33,38)(35,37)(36,44);(5,7,8,11)(6,10,9,12)(15,26)(20,35)(24,36)(25,28)(27,31)(34,37)(41,44)(42,43);(6,9)(10,12)(13,17,19,18)(14,22,23,33)(15,27,26,31)(16,29,30,32)(20,36,35,24)(21,38,40,39)(25,42,28,43)(34,41,37,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,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, 6427488);