#6822450

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1313 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)(10,14)(11,12)(17,18,23,28)(19,30,33,24)(20,34,37,25)(21,29,38,26)(22,39,32,27)(31,40,47,44)(35,41,46,48)(36,42,45,43);(9,10)(11,15)(12,16)(13,14)(17,19,23,33)(18,24,28,30)(20,35,37,46)(21,32,38,22)(25,41,34,48)(26,27,29,39)(31,45,47,36)(40,42,44,43);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20,21,36,23,37,38,45)(18,25,29,43,28,34,26,42)(19,31,22,35,33,47,32,46)(24,40,27,41,30,44,39,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21,23,38)(18,26,28,29)(19,32,33,22)(20,36,37,45)(24,27,30,39)(25,42,34,43)(31,46,47,35)(40,41,44,48);(10,14)(15,16)(17,22,38,19,23,32,21,33)(18,27,26,24,28,39,29,30)(20,35,45,31,37,46,36,47)(25,41,42,40,34,48,43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fhi,d^2=g,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6729646);