#483514

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3100	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

518 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)(3,5)(4,7)(6,8)(9,10,12,16,13,17,22,28,14,18,23,29,24,30,34,38)(11,15,20,26,21,27,33,37,19,25,31,35,32,36,39,40);(2,5)(3,8)(6,7);(9,11,12,20,13,21,22,33,14,19,23,31,24,32,34,39)(10,15,16,26,17,27,28,37,18,25,29,35,30,36,38,40);(11,19)(15,25)(20,31)(21,32)(26,35)(27,36)(33,39)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321307, #72322062, #72322081
maximal normal subgroup of	#72321307, #72322062, #72322081

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390710);