#6343669

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343394
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	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2757 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)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18)(19,27,30,23)(20,31)(21,25)(22,26)(24,33)(28,36,44,41)(29,42,46,37)(32,39)(34,38)(35,40)(43,47,45,48);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,22,30)(18,23,26,27)(20,28,33,44)(21,29,35,46)(24,36,31,41)(25,37,40,42)(32,43,34,45)(38,47,39,48);(2,7)(6,15)(8,12)(9,16)(17,20,22,33)(18,24,26,31)(19,28,30,44)(21,34,35,32)(23,36,27,41)(25,39,40,38)(29,45,46,43)(37,48,42,47);(17,21,22,35)(18,25,26,40)(19,29,30,46)(20,32,33,34)(23,37,27,42)(24,38,31,39)(28,43,44,45)(36,47,41,48);(18,26)(23,27)(24,31)(25,40)(36,41)(37,42)(38,39)(47,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=hi,f^2=i,b^a=bf,c^a=ch,e^a=ehi,f^a=fh,f^b=fi,g^b=ghi,d^c=dhi

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, 6250865);