#7612152

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1374 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,d^2=h,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,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, 7519348);