#10480920

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

4139 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,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, 10388116);