#6394045

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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#125 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2811 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18)(19,23)(20,24,31,37)(21,25)(22,26)(27,33,40,44)(28,34)(29,35)(30,36,42,46)(32,38)(39,43,47,48)(41,45);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19,22,29)(18,23,26,35)(20,27,31,40)(21,28,32,41)(24,33,37,44)(25,34,38,45)(30,39,42,47)(36,43,46,48);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20)(18,24,26,37)(19,27)(21,30)(22,31)(23,33,35,44)(25,36,38,46)(28,39)(29,40)(32,42)(34,43,45,48)(41,47);(3,12)(7,10)(11,13)(15,16)(17,21,22,32)(18,25,26,38)(19,28,29,41)(20,30,31,42)(23,34,35,45)(24,36,37,46)(27,39,40,47)(33,43,44,48);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(19,29)(23,35)(27,40)(28,41)(33,44)(34,45)(39,47)(43,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgh,d^2=i,e^2=f,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, 6301241);