#6343835

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	C₄²	C₂⁵
prime radical	C₂×#3518	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2799 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,22,28,23,27,21,26)(19,29,31,42,32,24,36,38)(20,25,33,39,34,40,35,41)(30,43,45,47,46,37,44,48);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,37)(26,38)(27,29)(28,42)(33,44)(34,46)(35,45)(39,47)(40,43)(41,48);(2,7)(6,14)(8,15)(9,16);(17,20,23,34)(18,25,27,40)(19,30,32,46)(21,35,22,33)(24,37,29,43)(26,41,28,39)(31,45,36,44)(38,48,42,47);(17,21,23,22)(18,26,27,28)(19,31,32,36)(20,33,34,35)(24,38,29,42)(25,39,40,41)(30,44,46,45)(37,47,43,48)

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=h,f^2=i,g^2=h,b^a=bf,c^a=ci,f^a=fh,f^b=fi,g^b=ghi,e^d=eh

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, 6251031);