#6458580

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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

1747 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)(9,10,13,15)(11,14,12,16)(17,18,23,29,24,30,22,28)(19,31,33,26,34,43,20,25)(21,36,39,45,40,27,38,46)(32,47,37,41,48,44,35,42);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,25)(20,23)(21,37)(22,33)(24,34)(26,30)(27,44)(28,31)(29,43)(32,39)(35,40)(36,47)(38,48)(41,46)(42,45);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20,22,34,24,33,23,19)(18,26,28,31,30,25,29,43)(21,35,38,48,40,37,39,32)(27,42,45,44,36,41,46,47);(1,4,5,3)(2,8,6,7)(17,21,23,39,24,40,22,38)(18,27,29,46,30,36,28,45)(19,32,33,37,34,48,20,35)(25,41,31,44,26,42,43,47);(17,22,24,23)(18,28,30,29)(19,33,34,20)(21,38,40,39)(25,31,26,43)(27,45,36,46)(32,37,48,35)(41,44,42,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365776);