#354404

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×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×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1212 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)(5,6,8,7)(9,10,14,21,16,22,15,20)(11,23,26,17,28,34,27,35)(12,29,13,19,31,36,30,18)(24,33,32,40,39,37,25,38);(9,11)(10,17,22,35)(12,24)(13,32)(14,26)(15,27)(16,28)(18,33,19,37)(20,23,21,34)(25,30)(29,40,36,38)(31,39);(2,4)(9,12,14,13,16,31,15,30)(10,18,20,36,22,19,21,29)(11,24,26,32,28,39,27,25)(17,33,23,38,35,37,34,40);(5,7,8,6)(9,13,14,31,16,30,15,12)(10,19,21,36,22,18,20,29)(11,25,26,24,28,32,27,39)(17,33,34,40,35,37,23,38)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,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, 261600);