#10529210

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	#51768	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3929 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18)(19,26)(20,29)(21,24)(22,28)(23,25)(27,31)(30,32)(33,34,38,41)(35,37,40,43)(36,39,46,48)(42,45,44,47);(1,2,5,8)(3,6,10,14)(4,7,12,16)(9,13,11,15)(17,19)(18,22)(20,30)(21,27)(23,28)(24,32)(25,26)(29,31)(33,35)(34,37)(36,44)(38,40)(39,47)(41,43)(42,46)(45,48);(17,20,23,21)(18,24,25,29)(19,27,28,30)(22,31,26,32)(33,36)(34,39)(35,42)(37,45)(38,46)(40,44)(41,48)(43,47);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,19)(18,22)(20,27)(21,30)(23,28)(24,31)(25,26)(29,32)(33,34)(35,43)(36,39)(37,40)(38,41)(42,47)(44,45)(46,48);(1,3,5,10)(2,6,8,14)(4,11,12,9)(7,15,16,13)(17,21,23,20)(18,24,25,29)(19,27,28,30)(22,32,26,31)(33,36)(34,39)(35,44)(37,47)(38,46)(40,42)(41,48)(43,45);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,22)(18,19)(20,31)(21,32)(23,26)(24,27)(25,28)(29,30)(33,37,38,43)(34,40,41,35)(36,45,46,47)(39,44,48,42)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,e^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, 10436406);