#10464567

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

4102 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18,21,24)(19,25,26,22)(20,23,30,29)(27,32,28,31)(33,34)(35,42)(36,41)(37,45)(38,44)(39,46)(40,47)(43,48);(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(19,26)(22,25)(27,28)(31,32)(33,35)(34,39)(36,43)(37,41)(38,46)(40,48)(42,44)(45,47);(17,19)(18,22)(20,27)(21,26)(23,31)(24,25)(28,30)(29,32)(33,36)(34,40)(35,43)(37,42)(38,47)(39,48)(41,44)(45,46);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,19)(18,22)(20,28)(21,26)(23,32)(24,25)(27,30)(29,31)(33,36)(34,41)(35,45)(37,42)(38,47)(39,48)(40,44)(43,46);(1,3,5,10)(2,6,8,14)(4,11,12,9)(7,15,16,13)(17,20,21,30)(18,23,24,29)(19,27,26,28)(22,31,25,32)(33,35)(34,42)(36,43)(37,40)(38,46)(39,44)(41,48)(45,47);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,18)(19,22)(20,29)(21,24)(23,30)(25,26)(27,32)(28,31)(33,37,38,48)(34,43,44,45)(35,41,46,40)(36,42,47,39)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,e^2=gh,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, 10371763);