#7610416

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1459 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)(17,18,23,28)(19,24,31,39)(20,25,30,38)(21,26,36,44)(22,27,34,42)(29,37,45,47)(32,40,33,41)(35,43,46,48);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,22)(21,33)(23,31)(25,27)(26,41)(28,39)(29,35)(30,34)(32,36)(37,43)(38,42)(40,44)(45,46)(47,48);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20)(18,25)(19,22)(21,32)(23,30)(24,27)(26,40)(28,38)(29,35)(31,34)(33,36)(37,43)(39,42)(41,44)(45,46)(47,48);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,21,34,46,23,36,22,35)(18,26,42,48,28,44,27,43)(19,29,20,32,31,45,30,33)(24,37,25,40,39,47,38,41);(2,8)(6,7)(17,22,23,34)(18,27,28,42)(19,20,31,30)(21,35,36,46)(24,25,39,38)(26,43,44,48)(29,32,45,33)(37,40,47,41)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=fhi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=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, 7517612);