#6393530

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×#3535	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3264 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303121, #72303498, #72303533
maximal normal subgroup of	#72303121, #72303498, #72303533

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6300726);