#93881

order	512
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	D₁₆
commutator	C₁₆	C₂×C₁₆
Frattini, cosocle	C₈×C₁₆	C₂²
socle, prime radical	C₂²	#1371
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈² ◁ C₈×C₁₆ ◁ G
LECS	G ▷ C₈×C₁₆ ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 2 generators on 48 points:

(1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,23)(19,28)(20,29)(21,32)(24,31)(25,40)(26,43)(27,37)(30,39)(33,46)(34,35)(36,41)(38,47)(42,48)(44,45);(17,18,24,38,20,25,30,36,32,41,45,40,46,47,19,23,22,27,28,35,33,42,44,43,21,26,39,48,29,34,31,37)

Polycyclic presentation:

2^9|a^2=d,b^2=cei,c^2=eg,d^2=f,e^2=gi,f^2=h,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gi

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1077);