#6393694

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342866
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×#540	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gh,d^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi,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, 6300890);