#10463934

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

4122 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)(17,18,22,25)(19,23,28,21)(20,24,30,32)(26,31,27,29)(34,39)(35,40)(37,45)(43,46);(1,3,7,5)(2,8,10,4)(6,11,14,13)(9,15,16,12)(17,19,22,28)(18,23,25,21)(20,27,30,26)(24,29,32,31)(34,39)(35,40)(41,47)(42,48);(1,3,7,5)(2,4,10,8)(6,11,14,13)(9,12,16,15)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,32)(27,28)(33,34,38,39)(35,36,40,44)(37,41,45,47)(42,43,48,46);(1,4,7,8)(2,3,10,5)(6,12,14,15)(9,11,16,13)(17,18)(19,21)(20,24)(22,25)(23,28)(26,29)(27,31)(30,32)(33,35)(34,36)(37,42)(38,40)(39,44)(41,43)(45,48)(46,47);(1,5,7,3)(2,8,10,4)(6,11,14,13)(9,12,16,15)(17,20)(18,24)(19,27)(21,31)(22,30)(23,29)(25,32)(26,28)(33,36)(34,40)(35,39)(37,46)(38,44)(41,42)(43,45)(47,48);(1,6,7,14)(2,9,10,16)(3,11,5,13)(4,12,8,15)(17,21)(18,19)(20,29)(22,23)(24,26)(25,28)(27,32)(30,31)(33,37)(34,41)(35,42)(36,43)(38,45)(39,47)(40,48)(44,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gh,d^2=g,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, 10371130);