#354305

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×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	C₂×#525	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1892 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)(9,10,14,18,15,19,13,29)(11,20,24,30,21,35,23,16)(12,17,27,33,28,34,26,40)(22,36,38,32,25,39,37,31);(2,4)(9,11)(10,16,19,30)(12,25)(13,23)(14,24)(15,21)(17,32,34,31)(18,20,29,35)(22,28)(26,38)(27,37)(33,39,40,36);(6,8)(11,21)(16,30)(20,35)(22,25)(23,24)(31,32)(36,39)(37,38);(9,12,14,27,15,28,13,26)(10,17,18,33,19,34,29,40)(11,22,24,38,21,25,23,37)(16,31,20,36,30,32,35,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,c^b=ci,d^b=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, 261501);