#93271

order	512
nilpotency class	4
isoclinism	isoclinic class of D₁₆
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	D₈
commutator	C₈	C₈²
Frattini, cosocle	C₄²×C₈	C₂²
socle	C₂²	#1277
prime radical	C₂³	#341
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂×C₄×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efi,f^a=fi,h^a=hi

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 467);