#6940132

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907869
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	#38035	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1525 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6847328);