#7509426

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

1279 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)(11,14)(12,16)(13,15)(17,18)(19,24)(20,31)(21,35)(22,27)(23,26)(25,34)(28,45)(29,41)(30,40)(32,48)(33,43)(36,39)(37,44)(38,47)(42,46);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,32)(21,29)(22,30)(23,36)(25,42)(26,39)(27,40)(28,33)(31,48)(34,46)(35,41)(37,47)(38,44)(43,45);(9,12,13,11)(10,14,15,16)(17,20)(18,25)(19,28)(21,33)(22,34)(23,37)(24,38)(26,43)(27,31)(29,46)(30,47)(32,36)(35,44)(39,42)(40,45)(41,48);(2,7)(3,8)(10,15)(14,16)(17,21,22,23)(18,26,27,35)(19,29,30,36)(20,33,34,37)(24,39,40,41)(25,43,31,44)(28,46,47,32)(38,42,45,48);(1,4,5,6)(2,3,7,8)(18,27)(24,40)(25,31)(26,35)(38,45)(39,41)(42,48)(43,44)

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,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, 7416622);