#6343625

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343406
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	C₂²×#41	D₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2798 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)(19,29,33,24)(20,25)(21,26)(22,27)(23,28)(30,43,45,37)(31,39,32,38)(34,40)(35,41)(36,42)(44,48,46,47);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,23,33)(18,24,28,29)(20,30,35,45)(21,31,22,32)(25,37,41,43)(26,38,27,39)(34,44,36,46)(40,47,42,48);(17,20,23,35)(18,25,28,41)(19,30,33,45)(21,36,22,34)(24,37,29,43)(26,42,27,40)(31,46,32,44)(38,48,39,47);(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,34,35,36)(24,38,29,39)(25,40,41,42)(30,44,45,46)(37,47,43,48);(2,7)(6,15)(8,12)(9,16)(17,22,23,21)(18,27,28,26)(19,32,33,31)(20,34,35,36)(24,39,29,38)(25,40,41,42)(30,44,45,46)(37,47,43,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=hi,e^2=hi,f^2=i,b^a=bf,e^a=ei,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, 6250821);