#10464575

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

4171 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)(3,4)(5,8)(6,9)(7,10)(11,12)(13,15)(14,16)(17,18,21,24)(19,25,26,22)(20,23,30,29)(27,32,28,31)(33,34)(35,36)(37,45)(38,39)(40,43)(41,47)(42,48)(44,46);(1,3,7,5)(2,8,10,4)(6,11,14,13)(9,15,16,12)(19,26)(22,25)(27,28)(31,32)(34,39)(36,40)(41,45)(44,48);(17,19)(18,22)(20,27)(21,26)(23,31)(24,25)(28,30)(29,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,48)(44,45)(46,47);(1,4,7,8)(2,3,10,5)(6,12,14,15)(9,11,16,13)(17,19)(18,22)(20,28)(21,26)(23,32)(24,25)(27,30)(29,31)(33,36,38,40)(34,35,39,43)(37,44,47,48)(41,42,45,46);(1,5,7,3)(2,8,10,4)(6,11,14,13)(9,12,16,15)(17,20,21,30)(18,23,24,29)(19,27,26,28)(22,31,25,32)(33,35)(34,36)(37,46)(38,43)(39,40)(41,48)(42,47)(44,45);(1,6,7,14)(2,9,10,16)(3,11,5,13)(4,12,8,15)(17,18)(19,22)(20,29)(21,24)(23,30)(25,26)(27,32)(28,31)(33,37,38,47)(34,41,39,45)(35,42,43,46)(36,44,40,48)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=g,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, 10371771);