#8750793

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2540	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=gi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8657989);