#6386114

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	C₂⁴	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3010 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)(19,23)(20,30,32,24)(21,25)(22,26)(27,41,43,35)(28,36)(29,37)(31,44,46,38)(33,39)(34,40)(42,48,45,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,29)(18,23,26,37)(20,27,32,43)(21,33,34,28)(24,35,30,41)(25,39,40,36)(31,45,46,42)(38,48,44,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,24)(19,27)(21,31)(22,32)(23,35)(25,38)(26,30)(28,42)(29,43)(33,45)(34,46)(36,47)(37,41)(39,48)(40,44);(3,11)(6,9)(10,16)(13,15);(2,8)(6,9)(7,14)(13,15)(17,21,22,34)(18,25,26,40)(19,28,29,33)(20,31,32,46)(23,36,37,39)(24,38,30,44)(27,42,43,45)(35,47,41,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,e^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, 6293310);