#10181220

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=i,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, 10088416);