#10366160

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

4107 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=ghi,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10273356);