#7507240

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1437 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)(17,18,21,25)(19,27,30,23)(20,24,31,37)(22,26,33,39)(28,40,34,35)(29,41,43,36)(32,38,44,47)(42,48,45,46);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(19,28)(22,33)(23,35)(26,39)(27,40)(29,42)(30,34)(32,44)(36,46)(38,47)(41,48)(43,45);(10,14)(13,16)(17,19,21,30)(18,23,25,27)(20,29,31,43)(22,34,33,28)(24,36,37,41)(26,40,39,35)(32,45,44,42)(38,48,47,46);(2,7)(3,8)(10,14)(13,16)(17,20)(18,24)(19,29)(21,31)(22,32)(23,36)(25,37)(26,38)(27,41)(28,42)(30,43)(33,44)(34,45)(35,46)(39,47)(40,48);(1,4,5,6)(2,3,7,8)(19,30)(20,31)(23,27)(24,37)(28,34)(32,44)(35,40)(38,47)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,h^b=hi,e^c=ei,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, 7414436);