#354352

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
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

1214 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=g,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,f^c=fi

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, 261548);