#10529819

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	#51567	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,e^2=h,f^2=hi,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, 10437015);