#391378

order	512
nilpotency class	3
isoclinism	isoclinic class of #219525
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2328
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

741 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298574);