#7501065

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35149	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2785 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=df,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, 7408261);