#10480790

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

4074 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=i,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, 10387986);