#6464953

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1375 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=hi,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,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, 6372149);