#7608745

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	#25125	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1660 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)(19,24)(20,25)(21,34)(22,27)(23,28)(26,36)(29,39)(30,41)(31,42)(32,40)(33,44)(35,43)(37,48)(38,46)(45,47);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,31)(18,24,28,42)(20,29,32,37)(21,33,36,47)(22,35,38,30)(25,39,40,48)(26,45,34,44)(27,43,46,41);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20,23,32)(18,25,28,40)(19,29,31,37)(21,35,36,30)(22,33,38,47)(24,39,42,48)(26,41,34,43)(27,44,46,45);(10,14)(13,16)(17,21,22,37,23,36,38,29)(18,26,27,39,28,34,46,48)(19,20,30,33,31,32,35,47)(24,40,41,45,42,25,43,44);(2,5)(6,8)(20,32)(21,36)(25,40)(26,34)(29,37)(33,47)(39,48)(44,45)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=h,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,e^d=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, 7515941);