#6467052

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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

1441 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)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,12,16)(17,18)(19,24)(20,25)(21,26)(22,28)(23,27)(29,38)(30,39)(31,41)(32,40)(33,42)(34,43)(35,44)(36,46)(37,45)(47,48);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,33)(21,29)(22,36)(23,32)(25,42)(26,38)(27,40)(28,46)(30,34)(31,37)(35,47)(39,43)(41,45)(44,48);(9,12,13,11)(10,14,15,16)(17,20,22,21,23,34,37,35)(18,25,28,26,27,43,45,44)(19,29,31,33,32,47,36,30)(24,38,41,42,40,48,46,39);(17,21,22,34,23,35,37,20)(18,26,28,43,27,44,45,25)(19,30,31,29,32,33,36,47)(24,39,41,38,40,42,46,48);(1,3,4,7)(2,6,5,8)(18,27)(20,34)(21,35)(24,40)(28,45)(29,47)(30,33)(41,46)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^c=ei,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, 6374248);