#10529206

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	#51815	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4117 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10436402);