#7590581

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1351 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=fi,c^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,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, 7497777);