#10247181

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	#46764	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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)(4,12)(7,9)(10,13)(17,18,23,27)(19,22,26,24)(20,25,21,30)(28,29,31,32)(33,34)(35,44)(36,46)(37,41)(38,45)(39,40)(42,47)(43,48);(1,2,5,6)(3,9,11,7)(4,8,12,14)(10,16,13,15)(17,19)(18,24)(20,28)(21,31)(22,27)(23,26)(25,32)(29,30)(34,40)(35,37)(42,46)(43,45);(1,2,5,6)(3,7,11,9)(4,8,12,14)(10,15,13,16)(17,20,23,21)(18,25,27,30)(19,28,26,31)(22,29,24,32)(33,35,39,37)(34,41,40,44)(36,43,47,45)(38,42,48,46);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,21,23,20)(18,25,27,30)(19,28,26,31)(22,32,24,29)(33,36,39,47)(34,42,40,46)(35,45,37,43)(38,44,48,41);(1,3,5,11)(2,7,6,9)(4,13,12,10)(8,16,14,15)(17,19)(18,22)(20,28)(21,31)(23,26)(24,27)(25,29)(30,32)(33,37,39,35)(34,41,40,44)(36,43,47,45)(38,46,48,42);(1,4)(2,8)(3,10)(5,12)(6,14)(7,15)(9,16)(11,13)(17,22)(18,26)(19,27)(20,29)(21,32)(23,24)(25,31)(28,30)(33,38)(34,43)(35,42)(36,44)(37,46)(39,48)(40,45)(41,47)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154377);