#10481063

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) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=i,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, 10388259);