#8734916

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

4044 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)(4,8)(5,7)(6,9)(11,16)(12,15)(13,14)(17,18)(19,23)(20,29)(21,25)(22,26)(24,28)(27,32)(30,31)(33,34,38,37)(35,40,43,39)(36,44,45,41)(42,46,48,47);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(17,19,22,27)(18,23,26,32)(20,21,28,31)(24,30,29,25)(33,35,38,43)(34,39,37,40)(36,42,45,48)(41,46,44,47);(4,12)(8,15)(11,13)(14,16)(19,27)(20,28)(23,32)(24,29)(33,35,38,43)(34,39,37,40)(36,42,45,48)(41,46,44,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20)(18,24)(19,21)(22,28)(23,30)(25,32)(26,29)(27,31)(33,35)(34,40)(36,42)(37,39)(38,43)(41,47)(44,46)(45,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,30)(22,26)(25,31)(27,32)(28,29)(33,36,38,45)(34,41,37,44)(35,42,43,48)(39,46,40,47);(1,5,6,3)(2,7,9,10)(4,11,12,13)(8,16,15,14)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,30)(29,32)(33,37,38,34)(35,40,43,39)(36,41,45,44)(42,46,48,47)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=h,e^2=h,f^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, 8642112);