#369366

order	512
nilpotency class	3
isoclinism	isoclinic class of #196152
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1398 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)(3,7)(4,6)(5,8)(9,10,14,18,13,17,20,29)(11,19,24,31,22,15,23,32)(12,25,28,16,26,35,27,34)(21,36,38,30,37,40,39,33);(1,3,5,4)(2,6,8,7)(9,11,20,23,13,22,14,24)(10,15,29,31,17,19,18,32)(12,21,27,39,26,37,28,38)(16,33,25,40,34,30,35,36);(1,3,5,4)(2,7,8,6)(9,12,14,27,13,26,20,28)(10,16,29,35,17,34,18,25)(11,21,24,39,22,37,23,38)(15,30,31,40,19,33,32,36);(1,4,5,3)(2,7,8,6)(10,17)(15,19)(16,34)(18,29)(25,35)(30,33)(31,32)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=efg,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=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, 276562);