#7507081

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1455 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)(9,10,12,14)(11,13,15,16)(17,18)(19,27)(20,24)(21,25)(22,26)(23,29)(28,45)(30,43)(31,48)(32,40)(33,41)(34,42)(35,38)(36,44)(37,46)(39,47);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(17,19,21,29)(18,23,25,27)(20,31,32,47)(22,35,34,30)(24,39,40,48)(26,43,42,38)(28,36,46,33)(37,44,45,41);(10,14)(13,16)(17,20,21,32)(18,24,25,40)(19,28,29,46)(22,36,34,33)(23,37,27,45)(26,44,42,41)(30,47,35,31)(38,48,43,39);(2,7)(3,8)(10,14)(13,16)(18,25)(19,29)(24,40)(26,42)(28,46)(30,35)(31,47)(41,44);(1,4,5,6)(2,3,7,8)(18,25)(20,32)(23,27)(26,42)(28,46)(31,47)(33,36)(38,43)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,e^c=ei,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, 7414277);