#7615360

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1367 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)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,32)(21,37)(22,27)(23,28)(25,35)(26,31)(29,42)(30,41)(33,40)(34,43)(36,47)(38,46)(39,45)(44,48);(1,3,5,4)(2,6,8,7)(11,16)(13,15)(17,19,22,30)(18,24,27,41)(20,33,35,29)(21,38,31,23)(25,42,32,40)(26,28,37,46)(34,48,39,36)(43,44,45,47);(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,39)(22,35)(23,36)(24,40)(26,45)(27,32)(28,44)(30,33)(31,34)(37,43)(38,48)(41,42)(46,47);(10,14)(13,15)(17,21,23,19,22,31,38,30)(18,26,28,41,27,37,46,24)(20,34,36,33,35,39,48,29)(25,43,44,40,32,45,47,42);(1,4,5,3)(2,7,8,6)(20,35)(25,32)(29,33)(34,39)(36,48)(40,42)(43,45)(44,47)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=gi,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

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, 7522556);