#9912951

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

4071 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)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,18)(19,24)(20,30)(21,26)(22,23)(25,31)(27,32)(28,29)(33,34,39,42)(35,40,38,48)(36,41,46,37)(43,44,45,47);(2,6)(7,13)(8,11)(14,15)(17,19,23,29)(18,24,22,28)(20,27,31,21)(25,26,30,32)(33,35,39,38)(34,40,42,48)(36,43,46,45)(37,47,41,44);(17,20,23,31)(18,25,22,30)(19,21,29,27)(24,32,28,26)(33,36,39,46)(34,37,42,41)(35,43,38,45)(40,47,48,44);(1,3,5,10)(2,7,6,13)(4,9,12,16)(8,14,11,15)(17,20)(18,25)(19,27)(21,29)(22,30)(23,31)(24,26)(28,32)(33,36)(34,41)(35,43)(37,42)(38,45)(39,46)(40,44)(47,48);(1,2)(3,7)(4,11)(5,6)(8,12)(9,15)(10,13)(14,16)(17,21,23,27)(18,26,22,32)(19,20,29,31)(24,30,28,25)(33,37,39,41)(34,36,42,46)(35,44,38,47)(40,45,48,43);(1,4)(2,8)(3,9)(5,12)(6,11)(7,14)(10,16)(13,15)(17,22,23,18)(19,28,29,24)(20,30,31,25)(21,26,27,32)(33,38,39,35)(34,40,42,48)(36,45,46,43)(37,47,41,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820147);