#10504571

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

4189 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)(3,10)(4,7)(5,8)(6,9)(11,15)(12,13)(14,16)(17,18,21,24)(19,22,27,25)(20,28,30,23)(26,32,29,31)(33,34)(35,36)(37,45)(38,39)(40,43)(41,47)(42,48)(44,46);(1,3,6,8)(2,5,9,10)(4,12,14,11)(7,15,16,13)(17,18,21,24)(19,22,27,25)(20,23,30,28)(26,31,29,32)(34,39)(36,40)(41,45)(44,48);(17,18)(19,22)(20,23)(21,24)(25,27)(26,31)(28,30)(29,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,48)(44,45)(46,47);(1,4,6,14)(2,7,9,16)(3,11,8,12)(5,13,10,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,32)(28,30)(29,31)(33,36,38,40)(34,35,39,43)(37,44,47,48)(41,42,45,46);(1,2)(3,5)(4,7)(6,9)(8,10)(11,13)(12,15)(14,16)(17,19,21,27)(18,22,24,25)(20,29,30,26)(23,32,28,31)(33,35)(34,36)(37,46)(38,43)(39,40)(41,48)(42,47)(44,45);(1,5)(2,8)(3,9)(4,13)(6,10)(7,12)(11,16)(14,15)(17,20)(18,23)(19,26)(21,30)(22,31)(24,28)(25,32)(27,29)(33,37)(34,41)(35,42)(36,44)(38,47)(39,45)(40,48)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=g,e^2=i,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, 10411767);