#7615400

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=gi,d^2=h,e^2=fi,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, 7522596);