#354913

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1566 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)(5,6,7,8)(9,10)(11,19)(12,16)(13,25)(14,18)(15,22)(17,26)(20,31)(21,32)(23,29)(24,37)(27,34)(28,35)(30,36)(33,39)(38,40);(9,11)(10,15,18,27)(12,21)(13,22)(14,20)(16,28,29,38)(17,19,25,31)(23,33)(24,35)(26,34)(30,32,37,39)(36,40);(2,4)(11,20)(12,23)(15,27)(16,29)(19,31)(22,34)(24,36)(30,37);(9,12,13,24,14,23,26,36)(10,16,25,37,18,29,17,30)(11,21,22,35,20,33,34,40)(15,28,31,39,27,38,19,32)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262109);