#10480985

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

4093 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)(3,8)(4,14)(5,15)(6,9)(7,13)(10,16)(11,12)(17,18)(19,20)(21,29)(22,23)(24,27)(25,31)(26,32)(28,30)(33,34)(35,44)(36,40)(37,41)(38,42)(39,43)(45,47)(46,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,9)(10,15)(14,16)(18,23)(20,24)(25,29)(28,32)(33,35,39,42)(34,38,43,44)(36,45,41,48)(37,47,40,46);(1,4)(2,5)(3,10)(6,15)(7,16)(8,12)(9,14)(11,13)(17,19)(18,24)(20,23)(21,26)(22,27)(25,32)(28,29)(30,31)(33,36,39,41)(34,40,43,37)(35,45,42,48)(38,46,44,47);(1,4)(2,5)(3,11)(6,15)(7,12)(8,16)(9,14)(10,13)(17,20,22,24)(18,19,23,27)(21,28,31,32)(25,26,29,30)(33,37,39,40)(34,41,43,36)(35,46,42,47)(38,45,44,48);(1,3)(2,8)(4,10)(5,12)(6,13)(7,9)(11,15)(14,16)(17,19)(18,20)(21,30)(22,27)(23,24)(25,32)(26,31)(28,29)(33,34)(35,38)(36,40)(37,41)(39,43)(42,44)(45,46)(47,48);(1,5)(2,4)(3,12)(6,14)(7,11)(8,10)(9,15)(13,16)(17,21)(18,25)(19,26)(20,28)(22,31)(23,29)(24,32)(27,30)(33,38)(34,42)(35,43)(36,46)(37,45)(39,44)(40,48)(41,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=gh,d^2=gi,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, 10388181);