#10521310

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

3989 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:

(1,2,7,9)(3,10,4,8)(5,6,15,16)(11,14,13,12)(17,18,22,25)(19,26,20,23)(21,29,30,24)(27,31,28,32)(33,34,39,42)(35,36,40,37)(38,41,46,48)(43,44,47,45);(17,18,22,25)(19,26,20,23)(21,24,30,29)(27,32,28,31)(33,35,39,40)(34,37,42,36)(38,43,46,47)(41,45,48,44);(1,3,7,4)(2,8,9,10)(5,11,15,13)(6,12,16,14)(17,19,22,20)(18,23,25,26)(21,27,30,28)(24,31,29,32)(33,36,39,37)(34,40,42,35)(38,44,46,45)(41,47,48,43);(1,4,7,3)(2,8,9,10)(5,11,15,13)(6,14,16,12)(17,18)(19,23)(20,26)(21,24)(22,25)(27,31)(28,32)(29,30)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,48)(46,47);(1,5)(2,6)(3,11)(4,13)(7,15)(8,12)(9,16)(10,14)(17,20,22,19)(18,23,25,26)(21,27,30,28)(24,32,29,31)(33,37,39,36)(34,35,42,40)(38,44,46,45)(41,47,48,43);(1,6)(2,5)(3,12)(4,14)(7,16)(8,11)(9,15)(10,13)(17,21)(18,24)(19,27)(20,28)(22,30)(23,31)(25,29)(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=gi,c^2=gh,d^2=hi,e^2=gi,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, 10428506);