#8096979

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

4103 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)(17,18)(19,27)(20,29)(21,23)(22,25)(24,26)(28,31)(30,32)(33,34,38,42)(35,39,44,47)(36,40,43,46)(37,41,45,48);(1,3,5,11)(2,6,9,15)(4,10,12,16)(7,13,8,14)(17,19,24,23)(18,21,26,27)(20,30,22,28)(25,32,29,31)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,9)(11,16)(14,15)(17,20,24,22)(18,25,26,29)(19,28,23,30)(21,31,27,32)(33,35,38,44)(34,39,42,47)(36,37,43,45)(40,41,46,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,14)(9,15)(12,16)(17,21)(18,19)(20,31)(22,32)(23,26)(24,27)(25,28)(29,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4)(2,8)(3,10)(5,12)(6,14)(7,9)(11,16)(13,15)(17,22,24,20)(18,25,26,29)(19,28,23,30)(21,32,27,31)(33,36)(34,40)(35,37)(38,43)(39,41)(42,46)(44,45)(47,48);(1,2)(3,6)(4,7)(5,9)(8,12)(10,13)(11,15)(14,16)(17,23,24,19)(18,21,26,27)(20,30,22,28)(25,31,29,32)(33,37)(34,41)(35,43)(36,44)(38,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,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, 8004175);