#7608706

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	#23854	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1663 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,25)(20,32)(21,38)(22,28)(23,27)(24,31)(26,36)(29,40)(30,41)(33,44)(34,43)(35,45)(37,47)(39,42)(46,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,33)(21,30)(22,31)(23,39)(25,28)(26,44)(27,41)(29,37)(32,46)(34,35)(36,48)(38,42)(40,45)(43,47);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,19)(18,25)(20,34)(21,39)(22,31)(23,30)(24,28)(26,40)(27,41)(29,36)(32,43)(33,35)(37,48)(38,42)(44,45)(46,47);(10,14)(13,16)(17,20,23,37,22,36,21,35)(18,26,27,45,28,32,38,47)(19,29,30,33,31,34,39,48)(24,40,42,44,25,43,41,46);(2,5)(6,8)(17,21,22,23)(18,27,28,38)(19,30,31,39)(20,35,36,37)(24,41,25,42)(26,45,32,47)(29,33,34,48)(40,46,43,44)

Polycyclic presentation:

2^9|c^2=fi,d^2=h,e^2=i,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, 7515902);