#483547

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	#466	C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

428 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390743);