#6393685

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	C₂⁴	C₂⁵
prime radical	C₂×#3553	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#125 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2826 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)(19,27)(20,24,32,39)(21,33)(22,26)(23,29)(25,35)(28,41,42,36)(30,37)(31,44,46,38)(34,40)(43,47,45,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,22,29)(18,23,26,27)(20,28,32,42)(21,34,35,30)(24,36,39,41)(25,40,33,37)(31,45,46,43)(38,48,44,47);(17,20)(18,24,26,39)(19,28)(21,31)(22,32)(23,36,27,41)(25,38,33,44)(29,42)(30,43)(34,45)(35,46)(37,47,40,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(19,29)(23,27)(28,42)(30,34)(36,41)(37,40)(43,45)(47,48);(2,8)(6,9)(7,16)(14,15)(17,21,22,35)(18,25,26,33)(19,30,29,34)(20,31,32,46)(23,37,27,40)(24,38,39,44)(28,43,42,45)(36,47,41,48)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gh,d^2=fi,e^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, 6300881);