#7505019

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1701 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:

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

Polycyclic presentation:

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

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, 7412215);