#7611275

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1517 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=hi,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, 7518471);