#6343613

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	#62793	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,d^2=hi,e^2=h,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, 6250809);