#6374344

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342659
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2922 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,e^b=ei,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6281540);