#8096830

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

4090 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:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=g,e^2=i,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, 8004026);