#7502979

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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	#30503	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1856 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)(13,14,18,23)(15,19,25,34)(16,27,29,37)(17,21,31,22)(20,30,35,28)(24,38,39,43)(26,36,41,33)(32,42,44,40);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,18,25)(14,19,23,34)(16,24,29,39)(17,26,31,41)(20,32,35,44)(21,36,22,33)(27,38,37,43)(28,40,30,42);(2,4)(13,16,18,29)(14,20,23,35)(15,24,25,39)(17,30,31,28)(19,32,34,44)(21,27,22,37)(26,42,41,40)(33,43,36,38);(6,11)(7,12)(14,21)(15,25)(17,31)(19,33)(20,27)(22,23)(24,39)(28,30)(32,43)(34,36)(35,37)(38,44);(5,8,9,10)(6,7,11,12)(15,25)(19,34)(24,39)(26,41)(32,44)(33,36)(38,43)(40,42)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^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,e^b=ei,g^c=gi,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, 7410175);