#7513703

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1340 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)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18,24,30)(19,25,35,43)(20,26,21,27)(22,28,23,29)(31,39,47,48)(32,40,36,44)(33,41,34,42)(37,45,38,46);(1,3,6,2)(4,7,5,8)(17,19)(18,25)(20,36)(21,32)(22,34)(23,33)(24,35)(26,44)(27,40)(28,42)(29,41)(30,43)(31,37)(38,47)(39,45)(46,48);(10,13)(14,16)(17,20,24,21)(18,26,30,27)(19,31,35,47)(22,37,23,38)(25,39,43,48)(28,45,29,46)(32,33,36,34)(40,41,44,42);(1,4,6,5)(2,7,3,8)(10,13)(14,16)(17,21,24,20)(18,27,30,26)(19,32,35,36)(22,37,23,38)(25,40,43,44)(28,45,29,46)(31,33,47,34)(39,41,48,42);(1,5,6,4)(2,7,3,8)(9,11,12,15)(10,14,13,16)(17,22,24,23)(18,28,30,29)(19,33,35,34)(20,37,21,38)(25,41,43,42)(26,45,27,46)(31,36,47,32)(39,44,48,40)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=fi,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,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, 7420899);