#6393857

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342866
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	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#3560	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3192 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=i,e^2=fi,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,g^c=gi,h^c=hi

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, 6301053);