#8260868

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	#39838	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi

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, 8168064);