#7502869

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500701
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	#30574	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1858 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,g^c=gi,e^d=ei,g^d=gi

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, 7410065);