#358716

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1436 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265912);