#8260870

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3926 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,e^2=i,f^2=gi,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, 8168066);