#6970823

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907977
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1355 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)(19,24)(20,31)(21,26)(22,36)(23,27)(25,32)(28,43)(29,38)(30,40)(33,41)(34,39)(35,42)(37,46)(44,47)(45,48);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19,23,30)(18,24,27,40)(20,28,33,45)(21,34,22,29)(25,37,42,47)(26,39,36,38)(31,43,41,48)(32,46,35,44);(3,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,20,21,35,23,33,22,32)(18,25,36,41,27,42,26,31)(19,28,34,44,30,45,29,46)(24,37,38,48,40,47,39,43);(2,6)(5,7)(10,13)(14,16)(17,21,23,22)(18,26,27,36)(19,29,30,34)(20,32,33,35)(24,38,40,39)(25,41,42,31)(28,44,45,46)(37,43,47,48);(9,11,12,15)(10,14,13,16)(20,33)(25,42)(28,45)(31,41)(32,35)(37,47)(43,48)(44,46)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,d^2=i,e^2=fh,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=dfhi,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6878019);