#483527

order	512
nilpotency class	3
isoclinism	isoclinic class of D₄×D₈
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

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	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

472 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321257, #72322062, #72322081
maximal normal subgroup of	#72321257, #72322062, #72322081

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390723);