#10480872

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

4340 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=i,e^2=g,f^2=g,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, 10388068);