#7505381

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1784 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7412577);