#6451665

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1553 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)(17,18,22,26)(19,23,29,38)(20,24,31,40)(21,25,35,44)(27,36,46,48)(28,37,33,42)(30,39,45,47)(32,41,34,43);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,19,22,29)(18,23,26,38)(20,30,31,45)(21,33,35,28)(24,39,40,47)(25,42,44,37)(27,34,46,32)(36,43,48,41);(10,15)(14,16)(17,20)(18,24)(19,27)(21,34)(22,31)(23,36)(25,43)(26,40)(28,45)(29,46)(30,33)(32,35)(37,47)(38,48)(39,42)(41,44);(9,12,13,11)(10,16,15,14)(19,28)(21,35)(23,37)(25,44)(27,45)(29,33)(30,46)(32,34)(36,47)(38,42)(39,48)(41,43);(1,3,4,7)(2,6,5,8)(19,29)(20,31)(23,38)(24,40)(28,33)(32,34)(37,42)(41,43)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi

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, 6358861);