#7504278

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500669
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35660	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2350 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27)(19,24,30,40)(20,25,33,44)(21,34,22,26)(28,37,46,47)(29,39,35,38)(31,41,45,48)(32,43,36,42);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,31)(21,29)(22,35)(23,30)(25,41)(26,38)(27,40)(28,32)(33,45)(34,39)(36,46)(37,43)(42,47)(44,48);(10,14)(13,16)(17,20,23,33)(18,25,27,44)(19,28,30,46)(21,32,22,36)(24,37,40,47)(26,42,34,43)(29,45,35,31)(38,41,39,48);(2,6)(3,8)(10,14)(13,16)(17,19)(18,24)(20,28)(21,35)(22,29)(23,30)(25,37)(26,39)(27,40)(31,32)(33,46)(34,38)(36,45)(41,43)(42,48)(44,47);(17,21,23,22)(18,26,27,34)(19,29,30,35)(20,32,33,36)(24,38,40,39)(25,42,44,43)(28,45,46,31)(37,41,47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=i,e^2=i,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,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, 7411474);