#6451366

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1820 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)(3,8)(4,6)(5,7)(9,10,13,14)(11,16,12,15)(17,18)(19,28)(20,26)(21,36)(22,25)(23,27)(24,32)(29,38)(30,45)(31,40)(33,48)(34,44)(35,41)(37,43)(39,47)(42,46);(1,3,5,4)(2,6,7,8)(10,14)(15,16)(17,19)(18,24)(20,33)(21,31)(22,32)(23,29)(25,28)(26,42)(27,40)(30,34)(35,46)(36,38)(37,47)(39,44)(41,48)(43,45);(1,4,5,3)(2,6,7,8)(9,11,13,12)(10,15,14,16)(18,25)(19,29)(21,23)(24,38)(26,41)(28,40)(30,33)(31,32)(34,37)(39,48)(42,45)(46,47);(9,12,13,11)(10,15,14,16)(17,20,22,35)(18,26,25,41)(19,30,32,47)(21,34,23,37)(24,39,28,45)(27,43,36,44)(29,33,31,46)(38,48,40,42);(17,21,22,23)(18,27,25,36)(19,31,32,29)(20,34,35,37)(24,40,28,38)(26,43,41,44)(30,46,47,33)(39,42,45,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=fg,c^2=fi,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,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, 6358562);