#6386129

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342800
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2918 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)(17,18,22,28,23,27,21,26)(19,24,30,38,31,39,35,41)(20,32,33,42,34,25,36,40)(29,43,45,47,46,37,44,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,23,31)(18,24,27,39)(20,29,34,46)(21,35,22,30)(25,37,32,43)(26,41,28,38)(33,45,36,44)(40,48,42,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,37)(26,40)(27,32)(28,42)(30,44)(31,46)(35,45)(38,47)(39,43)(41,48);(3,11)(6,9)(10,16)(13,15);(2,8)(6,9)(7,14)(13,15)(17,21,23,22)(18,26,27,28)(19,30,31,35)(20,33,34,36)(24,38,39,41)(25,40,32,42)(29,44,46,45)(37,47,43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,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, 6293325);