#10040710

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

4081 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=hi,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, 9947906);