#9937705

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

3956 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,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, 9844901);