#7613391

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1392 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)(19,26)(20,25)(21,34)(22,39)(23,29)(24,28)(27,37)(30,40)(31,41)(32,43)(33,42)(35,44)(36,45)(38,48)(46,47);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,23)(21,35)(22,31)(24,33)(26,29)(27,44)(28,41)(30,38)(32,36)(34,47)(37,46)(39,42)(40,45)(43,48);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20,23,19)(18,26,29,25)(21,30,37,32)(22,31,24,33)(27,40,34,43)(28,41,39,42)(35,36,46,38)(44,48,47,45);(10,14)(13,16)(17,21,22,36,23,37,24,38)(18,27,39,48,29,34,28,45)(19,30,33,46,20,32,31,35)(25,40,41,47,26,43,42,44);(1,4,5,3)(2,7,8,6)(17,22,23,24)(18,28,29,39)(19,31,20,33)(21,36,37,38)(25,41,26,42)(27,45,34,48)(30,35,32,46)(40,47,43,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=f,d^2=hi,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, 7520587);