#7611063

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1426 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)(4,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,28,31,24)(20,32,36,25)(21,38,23,26)(29,40,34,44)(30,41,39,42)(33,43,48,46)(35,45,37,47);(2,5)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,31)(18,24,27,28)(20,33,36,48)(21,30,23,39)(25,43,32,46)(26,41,38,42)(29,35,34,37)(40,47,44,45);(10,14)(11,15)(17,19)(18,24)(20,34)(21,39)(22,31)(23,30)(25,44)(26,42)(27,28)(29,36)(32,40)(33,35)(37,48)(38,41)(43,47)(45,46);(10,14)(13,16)(17,20,21,35,22,36,23,37)(18,25,38,47,27,32,26,45)(19,29,39,48,31,34,30,33)(24,40,41,46,28,44,42,43);(1,3,4,8)(2,6,5,7)(17,21,22,23)(18,26,27,38)(19,30,31,39)(20,35,36,37)(24,41,28,42)(25,45,32,47)(29,33,34,48)(40,46,44,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=hi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7518259);