#8734583

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

4004 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)(3,10)(4,6)(5,8)(7,9)(11,12)(13,15)(14,16)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31)(33,34,39,37)(35,40,36,43)(38,45,46,41)(42,48,44,47);(1,3,7,5)(2,8,9,10)(4,13,14,11)(6,12,16,15)(17,19,22,27)(18,23,26,32)(20,21,28,31)(24,30,29,25)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(2,9)(4,14)(8,10)(11,13)(19,27)(20,28)(23,32)(24,29)(33,36,39,35)(34,40,37,43)(38,44,46,42)(41,47,45,48);(1,4,7,14)(2,6,9,16)(3,11,5,13)(8,15,10,12)(17,20)(18,24)(19,21)(22,28)(23,30)(25,32)(26,29)(27,31)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,5,7,3)(2,10,9,8)(4,13,14,11)(6,12,16,15)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,32)(28,29)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48);(1,6)(2,4)(3,12)(5,15)(7,16)(8,13)(9,14)(10,11)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,30)(29,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8641779);