#6458597

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1779 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)(17,18,21,26,22,27,33,23)(19,24,28,37,30,38,42,34)(20,25,31,40,32,39,44,35)(29,36,41,46,43,47,48,45);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(18,23)(19,28)(20,31)(21,33)(26,27)(29,43)(30,42)(32,44)(34,37)(35,40)(36,46)(45,47);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,19,21,28,22,30,33,42)(18,24,26,37,27,38,23,34)(20,29,31,41,32,43,44,48)(25,36,40,46,39,47,35,45);(1,4,5,3)(2,8,6,7)(17,20,21,31,22,32,33,44)(18,25,26,40,27,39,23,35)(19,29,28,41,30,43,42,48)(24,36,37,46,38,47,34,45);(20,32)(25,39)(29,43)(31,44)(35,40)(36,47)(41,48)(45,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=fg,d^2=f,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^d=ei

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, 6365793);