#6818258

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1432 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)(9,10,12,14)(11,15,16,13)(19,27)(23,37)(28,43)(30,33)(31,38)(39,41)(44,46)(45,48);(2,5)(3,7)(17,18,21,25)(19,28,27,43)(20,32,34,47)(22,36,29,26)(23,38,37,31)(24,40,42,35)(30,41,33,39)(44,48,46,45);(9,11,12,16)(10,13,14,15)(17,19,29,38,21,27,22,31)(18,23,36,43,25,37,26,28)(20,33,40,45,34,30,35,48)(24,41,47,44,42,39,32,46);(2,5)(6,8)(19,27)(23,37)(28,43)(30,33)(31,38)(39,41)(44,46)(45,48);(1,3,4,7)(2,6,5,8)(17,20,29,40,21,34,22,35)(18,24,36,47,25,42,26,32)(19,30,38,48,27,33,31,45)(23,39,43,46,37,41,28,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=gh,e^2=fhi,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, 6725454);