#7509387

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1291 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,4)(5,7)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,24)(20,33)(21,27)(22,25)(23,28)(26,35)(29,42)(30,44)(31,37)(32,39)(34,41)(36,40)(38,45)(43,47)(46,48);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19,22,32)(18,24,25,39)(20,30,35,45)(21,31,23,29)(26,38,33,44)(27,37,28,42)(34,43,36,46)(40,48,41,47);(9,12,13,11)(10,14,15,16)(18,25)(19,29)(21,23)(24,37)(26,33)(30,43)(31,32)(34,36)(38,47)(39,42)(44,48)(45,46);(2,7)(3,8)(10,15)(14,16)(17,20,22,35)(18,26,25,33)(19,30,32,45)(21,36,23,34)(24,38,39,44)(27,41,28,40)(29,43,31,46)(37,47,42,48);(1,4,5,6)(2,3,7,8)(17,21,22,23)(18,27,25,28)(19,31,32,29)(20,34,35,36)(24,37,39,42)(26,40,33,41)(30,43,45,46)(38,48,44,47)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7416583);