#6393534

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×#3544	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3176 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:

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

Polycyclic presentation:

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

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, 6300730);