#8534791

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

4278 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gi,d^2=g,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8441987);