#7508420

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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1416 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)(11,15)(12,14)(13,16)(17,18,22,27)(19,28,32,24)(20,33,36,25)(21,37,23,26)(29,41,31,39)(30,40,34,42)(35,43,38,44)(45,47,46,48);(1,3)(2,6)(4,7)(5,8)(17,19,22,32)(18,24,27,28)(20,34,36,30)(21,31,23,29)(25,42,33,40)(26,39,37,41)(35,46,38,45)(43,47,44,48);(9,11,12,16)(10,13,14,15)(19,29)(21,23)(24,39)(26,37)(28,41)(30,45)(31,32)(34,46)(35,38)(40,47)(42,48)(43,44);(2,7)(3,8)(10,14)(13,15)(17,20,22,36)(18,25,27,33)(19,30,32,34)(21,38,23,35)(24,40,28,42)(26,44,37,43)(29,45,31,46)(39,47,41,48);(1,4,5,6)(2,3,7,8)(17,21,22,23)(18,26,27,37)(19,31,32,29)(20,35,36,38)(24,39,28,41)(25,43,33,44)(30,45,34,46)(40,48,42,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,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,e^c=ei,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, 7415616);