#7590504

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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

1360 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)(19,26)(20,25)(21,35)(22,37)(23,28)(24,30)(27,36)(29,40)(31,42)(32,39)(33,46)(34,47)(38,44)(41,45)(43,48);(3,8)(5,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,30,23,31,36,21)(18,24,27,26,28,35,37,42)(20,29,34,44,32,43,45,33)(25,38,41,40,39,46,47,48);(1,3,4,8)(2,5,6,7)(17,20)(18,25)(19,29)(21,33)(22,34)(23,32)(24,38)(26,40)(27,41)(28,39)(30,44)(31,43)(35,46)(36,45)(37,47)(42,48);(2,6)(5,7)(11,15)(13,16)(20,32)(25,39)(29,43)(33,44)(34,45)(38,46)(40,48)(41,47);(17,21,36,31,23,30,22,19)(18,26,37,24,28,42,27,35)(20,33,45,43,32,44,34,29)(25,40,47,38,39,48,41,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=g,e^2=fi,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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7497700);