#354636

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

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	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1598 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)(6,8)(9,10,15,19,16,21,14,20)(11,22,26,35,27,34,25,17)(12,28,13,18,30,36,29,37)(23,32,31,38,40,33,24,39);(2,4)(9,11)(10,17,21,35)(12,23)(13,31)(14,25)(15,26)(16,27)(18,32,37,33)(19,22,20,34)(24,29)(28,39,36,38)(30,40);(5,6,7,8)(9,12,15,13,16,30,14,29)(10,18,20,28,21,37,19,36)(11,23,26,31,27,40,25,24)(17,32,34,39,35,33,22,38);(9,13,15,30,16,29,14,12)(10,18,19,36,21,37,20,28)(11,24,26,23,27,31,25,40)(17,33,22,39,35,32,34,38)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,d^b=di,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, 261832);