#7505101

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1805 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)(3,5)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,25)(19,23,28,38)(20,24,32,40)(21,26,35,42)(27,36,45,47)(29,37,33,43)(30,39,46,48)(31,41,34,44);(1,3)(2,4)(5,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,30)(21,33)(22,28)(24,39)(25,38)(26,43)(27,31)(29,35)(32,46)(34,45)(36,41)(37,42)(40,48)(44,47);(1,4,6,5)(2,7,8,3)(10,14)(13,16)(17,20,22,32)(18,24,25,40)(19,27,28,45)(21,34,35,31)(23,36,38,47)(26,44,42,41)(29,46,33,30)(37,48,43,39);(1,5,6,4)(2,7,8,3)(10,14)(13,16);(18,25)(19,28)(24,40)(26,42)(27,45)(29,33)(30,46)(41,44)

Polycyclic presentation:

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

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, 7412297);