#6457890

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1518 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)(11,13,15,16)(17,18)(19,23)(20,32)(21,25)(22,27)(24,36)(26,34)(28,44)(29,38)(30,39)(31,43)(33,47)(35,42)(37,46)(40,45)(41,48);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,33)(21,31)(22,29)(24,40)(25,43)(26,39)(27,38)(28,35)(30,34)(32,47)(36,45)(37,41)(42,44)(46,48);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,20,34,48,22,35,21,36)(18,24,25,42,27,41,26,32)(19,28,30,45,29,33,31,46)(23,37,43,47,38,40,39,44);(1,4,5,3)(2,6,8,7)(11,15)(13,16)(19,29)(20,35)(23,38)(24,41)(30,31)(32,42)(36,48)(39,43);(17,21,22,34)(18,25,27,26)(19,30,29,31)(20,36,35,48)(23,39,38,43)(24,42,41,32)(28,45,33,46)(37,44,40,47)

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6365086);