#6465302

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1358 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)(3,7)(4,6)(5,8)(17,18,24,30)(19,26,33,41)(20,25,32,42)(21,27,37,46)(22,29,23,28)(31,43,35,45)(34,40,36,44)(38,48,39,47);(1,3,5,4)(2,6,8,7)(9,10)(11,15)(12,13)(14,16)(17,19,22,20,24,33,23,32)(18,25,28,26,30,42,29,41)(21,35,38,36,37,31,39,34)(27,44,47,45,46,40,48,43);(1,3,5,4)(2,7,8,6)(10,13)(14,15)(17,20,22,33,24,32,23,19)(18,26,28,42,30,41,29,25)(21,36,38,31,37,34,39,35)(27,45,47,40,46,43,48,44);(9,11,12,16)(10,14,13,15)(17,21)(18,27)(19,31)(20,34)(22,38)(23,39)(24,37)(25,40)(26,43)(28,47)(29,48)(30,46)(32,36)(33,35)(41,45)(42,44);(1,4,5,3)(2,6,8,7)(21,37)(27,46)(31,35)(34,36)(38,39)(40,44)(43,45)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=hi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,d^c=di,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372498);