#7610797

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1463 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)(17,18,22,26)(19,23,28,36)(20,30,33,24)(21,25,34,40)(27,39,32,35)(29,37,42,47)(31,43,45,38)(41,48,44,46);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(20,31)(21,34)(24,38)(25,40)(27,41)(29,42)(30,43)(32,44)(33,45)(35,46)(37,47)(39,48);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,19)(18,23)(20,32)(21,29)(22,28)(24,39)(25,37)(26,36)(27,33)(30,35)(31,44)(34,42)(38,48)(40,47)(41,45)(43,46);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,20,21,31,22,33,34,45)(18,24,25,38,26,30,40,43)(19,27,29,41,28,32,42,44)(23,35,37,46,36,39,47,48);(2,8)(6,7)(19,28)(23,36)(27,32)(29,42)(35,39)(37,47)(41,44)(46,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dgi,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, 7517993);