#387069

order	512
nilpotency class	3
isoclinism	isoclinic class of #213752
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂⁴	C₂³×C₄
Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=eg,d^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^b=hi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 294265);