#10181327

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3850 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=i,e^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, 10088523);