#7505197

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500695
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1520 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)(3,6,4,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,26)(21,27)(22,24)(23,28)(29,40)(30,39)(31,36)(32,37)(33,41)(34,42)(35,38)(43,48)(44,47)(45,46);(1,2)(3,6)(4,8)(5,7)(9,11,12,15)(10,13,14,16)(18,24)(19,29)(20,32)(21,23)(25,38)(30,44)(31,35)(36,40)(39,48)(41,42)(43,45)(46,47);(1,3,5,4)(2,6,7,8)(10,14)(13,16)(17,19)(18,25)(20,30)(21,29)(22,31)(23,35)(24,36)(26,39)(27,40)(28,38)(32,45)(33,44)(34,43)(37,46)(41,47)(42,48);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,22,32)(18,26,24,37)(19,30,31,45)(21,34,23,33)(25,39,36,46)(27,42,28,41)(29,43,35,44)(38,47,40,48);(17,21,22,23)(18,27,24,28)(19,29,31,35)(20,33,32,34)(25,40,36,38)(26,41,37,42)(30,44,45,43)(39,47,46,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,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, 7412393);