#6458347

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1650 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18,22,27,24,29,23,28)(19,30,20,25,33,39,32,40)(21,26,35,42,37,44,36,43)(31,45,34,38,47,41,46,48);(2,6)(7,8)(17,19)(18,25)(20,23)(21,34)(22,32)(24,33)(26,41)(27,30)(28,39)(29,40)(31,35)(36,47)(37,46)(38,42)(43,48)(44,45);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20,23,19,24,32,22,33)(18,25,28,30,29,40,27,39)(21,34,36,31,37,46,35,47)(26,38,43,45,44,48,42,41);(1,4,5,3)(2,8,6,7)(17,21,23,36,24,37,22,35)(18,26,28,43,29,44,27,42)(19,31,32,46,33,47,20,34)(25,38,30,45,40,48,39,41);(17,22,24,23)(18,27,29,28)(19,32,33,20)(21,35,37,36)(25,30,40,39)(26,42,44,43)(31,46,47,34)(38,45,48,41)

Polycyclic presentation:

2^9|a^2=gh,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365543);