#6386377

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342800
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×#3521	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3089 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)(12,23,25,16)(13,17)(14,18)(15,22)(20,28,35,33)(21,31)(24,36,38,30)(26,29)(27,32)(34,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,26)(14,22)(16,28)(17,31)(18,19)(21,27)(23,33)(24,37)(25,35)(29,32)(30,40)(34,38)(36,39);(9,12)(10,16)(11,20)(13,24)(14,25)(15,28)(17,30)(18,23)(19,33)(21,34)(22,35)(26,37)(27,38)(29,39)(31,40)(32,36);(2,5)(6,8)(9,13,14,27)(10,17,18,32)(11,21,22,26)(12,24,25,38)(15,29,19,31)(16,30,23,36)(20,34,35,37)(28,39,33,40);(1,3,4,7)(2,6,5,8)(10,18)(11,22)(16,23)(17,32)(20,35)(21,26)(30,36)(34,37)

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=fi,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,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, 6293573);