#359570

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3486	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3826 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)(9,10)(11,16)(12,22,24,34)(13,26)(14,18)(15,19)(17,25,32,28)(20,36,23,39)(21,31)(27,33)(29,35)(30,37,40,38);(2,4)(5,6)(7,8)(9,11,15,13)(10,16,19,26)(12,20,24,23)(14,27,29,21)(17,30,32,40)(18,33,35,31)(22,39,34,36)(25,38,28,37);(6,8)(9,12)(10,17)(11,20)(13,23)(14,28)(15,24)(16,30)(18,34)(19,32)(21,38)(22,35)(25,29)(26,40)(27,37)(31,39)(33,36);(2,4)(9,13,15,11)(10,16,19,26)(12,23,24,20)(14,21,29,27)(17,30,32,40)(18,33,35,31)(22,39,34,36)(25,37,28,38)

Polycyclic presentation:

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

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, 266766);