#8096992

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

4170 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)(3,6)(4,8)(5,9)(7,12)(10,14)(11,15)(13,16)(18,24)(20,25)(26,31)(28,32)(33,34,39,42)(35,37,44,40)(36,45,38,41)(43,48,46,47);(1,3,5,11)(2,6,9,15)(4,10,12,16)(7,13,8,14)(17,18,23,24)(19,25,21,20)(22,26,30,31)(27,32,29,28)(33,35)(34,40)(36,46)(37,42)(38,43)(39,44)(41,47)(45,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,9)(11,16)(14,15)(17,19,23,21)(18,25,24,20)(22,27,30,29)(26,32,31,28)(33,36,39,38)(34,41,42,45)(35,43,44,46)(37,47,40,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,14)(9,15)(12,16)(17,20)(18,19)(21,24)(22,28)(23,25)(26,27)(29,31)(30,32)(33,37)(34,35)(36,47)(38,48)(39,40)(41,43)(42,44)(45,46);(1,4)(2,8)(3,10)(5,12)(6,14)(7,9)(11,16)(13,15)(17,21,23,19)(18,20,24,25)(22,27,30,29)(26,32,31,28)(33,35)(34,37)(36,43)(38,46)(39,44)(40,42)(41,47)(45,48);(1,2)(3,6)(4,7)(5,9)(8,12)(10,13)(11,15)(14,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,38,39,36)(34,41,42,45)(35,43,44,46)(37,48,40,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^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, 8004188);