#10216500

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	#59674	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3974 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=hi,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10123696);