#484126

order	512
nilpotency class	4
isoclinism	isoclinic class of #33830
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

417 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391322);