#7156949

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996375
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1925 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei

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, 7064145);