#354062

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2798 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)(3,4)(6,8)(9,10,13,19,15,18,14,27)(11,16,22,30,21,29,25,31)(12,17,24,32,23,34,26,33)(20,35,37,40,36,28,38,39);(2,4)(9,11,14,22,15,21,13,25)(10,16,19,31,18,29,27,30)(12,20,24,38,23,36,26,37)(17,28,33,39,34,35,32,40);(5,6,7,8)(9,12,14,24,15,23,13,26)(10,17,19,33,18,34,27,32)(11,20,22,38,21,36,25,37)(16,28,31,39,29,35,30,40);(11,21)(16,29)(20,36)(22,25)(28,35)(30,31)(37,38)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261258);