#354100

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷×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	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2858 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:

(2,4)(9,10,14,19,15,18,13,27)(11,20,22,16,23,31,28,30)(12,17,26,32,24,33,25,37)(21,34,36,29,35,39,38,40);(5,6,7,8)(9,11)(10,16)(12,21)(13,28)(14,22)(15,23)(17,29)(18,30)(19,31)(20,27)(24,35)(25,38)(26,36)(32,39)(33,40)(34,37);(1,2,3,4)(9,12,14,26,15,24,13,25)(10,17,19,32,18,33,27,37)(11,21,22,36,23,35,28,38)(16,29,31,39,30,40,20,34);(10,18)(12,24)(16,30)(19,27)(20,31)(21,35)(25,26)(36,38)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,e^b=ei,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, 261296);