#7511911

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	#35210	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1627 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)(19,28,31,24)(20,25,35,42)(21,36,23,26)(29,45,46,38)(30,39,32,44)(33,48,47,40)(34,43,37,41);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,27,28)(20,33,35,47)(21,32,23,30)(25,40,42,48)(26,44,36,39)(29,37,46,34)(38,41,45,43);(10,14)(13,15)(17,20,22,35)(18,25,27,42)(19,29,31,46)(21,34,23,37)(24,38,28,45)(26,41,36,43)(30,33,32,47)(39,48,44,40);(2,7)(3,8)(11,16)(13,15)(17,21,22,23)(18,26,27,36)(19,30,31,32)(20,34,35,37)(24,39,28,44)(25,41,42,43)(29,33,46,47)(38,48,45,40);(1,4,5,6)(2,3,7,8)(18,27)(24,28)(25,42)(26,36)(38,45)(39,44)(40,48)(41,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=fgi,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, 7419107);