#7609783

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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1752 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,30)(21,23)(22,26)(25,33)(27,40)(28,35)(29,38)(31,39)(32,37)(34,36)(41,45)(42,47)(43,46)(44,48);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(18,23)(20,31)(22,34)(24,35)(25,39)(27,42)(29,43)(30,41)(32,44)(33,45)(37,47)(40,48);(10,14)(11,15)(17,19)(18,24)(20,32)(21,28)(22,29)(23,35)(25,40)(26,38)(27,33)(30,37)(31,44)(34,43)(36,46)(39,48)(41,47)(42,45);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,22,31,21,33,34,45)(18,25,26,41,23,30,36,39)(19,27,29,42,28,32,43,44)(24,37,38,48,35,40,46,47);(2,7)(6,8)(18,23)(19,28)(25,30)(26,36)(27,32)(29,43)(39,41)(42,44)

Polycyclic presentation:

2^9|b^2=fi,d^2=fhi,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, 7516979);