#7507296

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

1452 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)(19,23)(20,30)(21,26)(22,25)(24,31)(27,35)(28,42)(29,37)(32,44)(33,39)(34,40)(36,43)(38,45)(41,48)(46,47);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(19,27)(21,33)(23,35)(26,39)(28,41)(29,34)(32,45)(36,47)(37,40)(38,44)(42,48)(43,46);(10,14)(13,16)(17,19,22,29)(18,23,25,37)(20,28,31,43)(21,34,33,27)(24,36,30,42)(26,40,39,35)(32,46,45,41)(38,48,44,47);(2,7)(3,8)(10,14)(13,16)(17,20)(18,24)(19,28)(21,32)(22,31)(23,36)(25,30)(26,38)(27,41)(29,43)(33,45)(34,46)(35,47)(37,42)(39,44)(40,48);(1,4,5,6)(2,3,7,8)(18,25)(20,31)(23,37)(26,39)(28,43)(32,45)(35,40)(41,46)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,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, 7414492);