#484084

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂×#508	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

455 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,d^a=dhi,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=dhi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391280);