#8096876

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

4182 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004072);