#354765

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1162 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)(3,8)(4,6)(5,7)(9,10,13,17,14,18,25,31)(11,19,22,33,21,15,26,28)(12,16,24,30,23,29,36,39)(20,32,35,40,34,27,37,38);(9,11)(10,15,18,19)(12,20)(13,22)(14,21)(16,27,29,32)(17,28,31,33)(23,34)(24,35)(25,26)(30,38,39,40)(36,37);(1,3,5,4)(2,6,7,8)(12,23)(16,29)(20,34)(24,36)(27,32)(30,39)(35,37)(38,40);(1,4,5,3)(2,6,7,8)(9,12,13,24,14,23,25,36)(10,16,17,30,18,29,31,39)(11,20,22,35,21,34,26,37)(15,27,28,38,19,32,33,40)

Polycyclic presentation:

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

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, 261961);