#6343320

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	#62805	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3164 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:

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

Polycyclic presentation:

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

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, 6250516);