#6833112

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1242 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=f,e^2=ghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6740308);