#8097057

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

4192 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,e^2=gh,f^2=g,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, 8004253);