#10482506

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

4161 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10389702);