#363435

order	512
nilpotency class	3
isoclinism	isoclinic class of #214672
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	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

1247 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)(3,7)(4,5)(6,8)(9,10)(11,19,22,15)(12,23)(13,26)(14,17)(16,24)(18,29)(20,36,38,40)(21,31,27,34)(25,35)(28,33)(30,37,32,39);(2,5)(6,7)(9,11)(10,15)(12,20)(13,27)(14,22)(16,32)(17,19)(18,34)(21,29)(23,40)(24,39)(25,38)(26,31)(28,37)(30,33)(35,36);(9,12,14,25)(10,16,17,33)(11,20,22,38)(13,28,29,24)(15,30,19,32)(18,35,26,23)(21,39,27,37)(31,36,34,40);(1,3,4,8)(2,6,5,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,30)(17,19)(18,31)(23,40)(24,37)(25,38)(26,34)(27,29)(28,39)(32,33)(35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270631);