#10455678

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

3980 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=h,e^2=gh,f^2=h,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, 10362874);