#362827

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

1244 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)(4,6)(5,8)(9,10)(11,16,22,33)(12,24,13,18)(14,20)(15,19)(17,26,35,27)(21,31)(23,32,29,36)(25,38)(28,34)(30,39)(37,40);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,25)(13,21)(14,22)(15,29)(17,30)(18,31)(19,36)(20,33)(23,28)(24,38)(26,37)(27,39)(32,34)(35,40);(9,12,14,13)(10,17)(11,21,22,25)(15,26,28,27)(16,30)(18,34)(19,24)(20,35)(23,37,29,39)(31,32)(33,40)(36,38);(2,6)(5,7)(9,13,14,12)(10,18,20,24)(11,21,22,25)(15,27,28,26)(16,31,33,38)(17,34,35,19)(23,37,29,39)(30,32,40,36)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=g,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,h^b=hi,h^c=hi,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, 270023);