#6343648

order	512
nilpotency class	3
isoclinism	isoclinic class of #6343312
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

2973 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6250844);