#484085

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	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

463 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,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, 391281);