#10040465

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
prime radical	#57307	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3951 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)(3,7)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,23,21)(19,27,29,24)(20,30,31,25)(22,32,28,26)(34,37)(39,41)(40,44)(46,47);(1,3,6,11)(2,7,8,16)(4,14,10,12)(5,13,9,15)(17,19,23,29)(18,24,21,27)(20,28,31,22)(25,26,30,32)(33,34,38,37)(35,41,43,39)(36,44,45,40)(42,46,48,47);(2,8)(3,11)(4,10)(13,15)(17,20,23,31)(18,25,21,30)(19,22,29,28)(24,32,27,26)(33,34,38,37)(35,41,43,39)(36,44,45,40)(42,46,48,47);(1,4,6,10)(2,9,8,5)(3,12,11,14)(7,13,16,15)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,26)(27,32)(33,35,38,43)(34,39,37,41)(36,42,45,48)(40,46,44,47);(1,3)(2,7)(4,12)(5,15)(6,11)(8,16)(9,13)(10,14)(17,21,23,18)(19,24,29,27)(20,30,31,25)(22,32,28,26)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,5,6,9)(2,10,8,4)(3,13,11,15)(7,12,16,14)(17,22,23,28)(18,26,21,32)(19,20,29,31)(24,30,27,25)(33,37,38,34)(35,41,43,39)(36,40,45,44)(42,46,48,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9947661);