#7509398

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500689
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

1293 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)(17,18,22,27)(19,29,33,24)(20,34,37,25)(21,26,23,28)(30,41,32,39)(31,40,35,42)(36,44,38,43)(45,47,46,48);(1,3)(2,6)(4,7)(5,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,35)(21,32)(22,33)(23,30)(25,42)(26,41)(27,29)(28,39)(31,37)(34,40)(36,46)(38,45)(43,48)(44,47);(9,12,13,11)(10,14,15,16)(19,30)(21,23)(24,39)(26,28)(29,41)(31,45)(32,33)(35,46)(36,38)(40,47)(42,48)(43,44);(2,7)(3,8)(10,15)(14,16)(17,20,22,37)(18,25,27,34)(19,31,33,35)(21,38,23,36)(24,40,29,42)(26,44,28,43)(30,45,32,46)(39,47,41,48);(1,4,5,6)(2,3,7,8)(17,21,22,23)(18,26,27,28)(19,32,33,30)(20,36,37,38)(24,41,29,39)(25,43,34,44)(31,45,35,46)(40,47,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,e^c=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7416594);