#483382

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	C₂²×C₄⋊C₈
prime radical	C₂×#41	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

475 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231976, #72231978, #72231979
maximal normal subgroup of	#72231976, #72231978, #72231979

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390578);