#6393545

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	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3134 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)(9,10,14,20,15,19,13,18)(11,16,22,29,23,30,26,33)(12,17,24,31,25,32,27,34)(21,28,37,39,36,40,35,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,21)(13,26)(14,22)(15,23)(17,28)(18,33)(19,30)(20,29)(24,37)(25,36)(27,35)(31,39)(32,40)(34,38);(9,12,13,24,15,25,14,27)(10,17,20,34,19,32,18,31)(11,21,26,37,23,36,22,35)(16,28,29,38,30,40,33,39);(3,8)(5,7);(2,6)(5,7)(9,13,15,14)(10,18,19,20)(11,22,23,26)(12,24,25,27)(16,29,30,33)(17,31,32,34)(21,35,36,37)(28,38,40,39)

Polycyclic presentation:

2^9|a^2=h,c^2=ghi,e^2=i,h^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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6300741);