#6820244

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	C₂×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1385 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)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(17,18)(19,24)(20,34)(21,26)(22,27)(23,28)(25,37)(29,48)(30,41)(31,42)(32,43)(33,44)(35,46)(36,47)(38,40)(39,45);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,23,32)(18,24,28,43)(20,22,37,39)(21,30,35,33)(25,45,34,27)(26,41,46,44)(29,36,38,31)(40,42,48,47);(9,12,13,11)(10,14,15,16)(17,20,35,36,23,37,21,31)(18,25,46,42,28,34,26,47)(19,29,30,39,32,38,33,22)(24,40,41,27,43,48,44,45);(2,6)(5,7)(17,21,23,35)(18,26,28,46)(19,30,32,33)(20,36,37,31)(22,38,39,29)(24,41,43,44)(25,42,34,47)(27,40,45,48);(3,8)(5,7)(17,22,35,29,23,39,21,38)(18,27,46,48,28,45,26,40)(19,31,30,20,32,36,33,37)(24,42,41,34,43,47,44,25)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=i,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6727440);