#354634

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
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×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1590 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)(9,10,15,20,16,21,14,32)(11,22,26,34,27,38,25,17)(12,28,13,19,30,36,29,18)(23,35,31,40,39,33,24,37);(2,4)(9,11)(10,17,21,34)(12,23)(13,31)(14,25)(15,26)(16,27)(18,33,19,35)(20,22,32,38)(24,29)(28,37,36,40)(30,39);(5,6,7,8)(9,12,14,29,16,30,15,13)(10,18,20,28,21,19,32,36)(11,23,25,24,27,39,26,31)(17,33,22,37,34,35,38,40);(9,13,15,30,16,29,14,12)(10,19,20,36,21,18,32,28)(11,24,26,23,27,31,25,39)(17,33,22,37,34,35,38,40)

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

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, 261830);