#9814607

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

4133 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,23,25)(19,27,21,24)(20,22,30,26)(28,32,31,29)(33,34)(35,37)(36,41)(38,47)(39,40)(42,44)(43,46)(45,48);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,23,21)(18,24,25,27)(20,28,30,31)(22,29,26,32)(33,35)(34,37)(36,44)(38,45)(39,43)(40,46)(41,42)(47,48);(2,3)(7,9)(8,10)(14,15)(18,25)(20,30)(24,27)(28,31)(34,40)(35,43)(36,45)(42,47);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,26)(19,28)(21,31)(22,25)(23,30)(24,32)(27,29)(33,36,39,45)(34,41,40,48)(35,44,43,38)(37,42,46,47);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,23,19)(18,24,25,27)(20,31,30,28)(22,29,26,32)(33,37)(34,35)(36,42)(38,48)(39,46)(40,43)(41,44)(45,47);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,22,23,26)(18,20,25,30)(19,29,21,32)(24,28,27,31)(33,38,39,44)(34,42,40,47)(35,45,43,36)(37,41,46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9721803);