#10247310

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	#53431	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4166 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)(4,5)(6,8)(7,14)(9,11)(12,13)(15,16)(17,18)(19,20)(21,29)(22,24)(23,31)(25,27)(26,30)(28,32)(33,34)(35,41)(36,38)(37,43)(39,40)(42,46)(44,45)(47,48);(1,3,6,14)(2,7,8,10)(4,15,11,12)(5,13,9,16)(17,19)(18,20)(21,26)(22,27)(23,28)(24,25)(29,30)(31,32)(34,40)(36,46)(41,43)(44,47);(2,8)(4,11)(7,10)(12,15)(17,18,22,24)(19,25,27,20)(21,23,31,29)(26,30,32,28)(33,35,39,37)(34,41,40,43)(36,44,46,47)(38,45,42,48);(1,4,6,11)(2,9,8,5)(3,12,14,15)(7,13,10,16)(17,18)(19,25)(20,27)(21,23)(22,24)(26,30)(28,32)(29,31)(33,36)(34,42)(35,44)(37,47)(38,40)(39,46)(41,48)(43,45);(1,3)(2,10)(4,15)(5,16)(6,14)(7,8)(9,13)(11,12)(17,20,22,25)(18,19,24,27)(21,30,31,28)(23,32,29,26)(33,37,39,35)(34,43,40,41)(36,44,46,47)(38,45,42,48);(1,5,6,9)(2,11,8,4)(3,13,14,16)(7,12,10,15)(17,21)(18,23)(19,26)(20,28)(22,31)(24,29)(25,30)(27,32)(33,38,39,42)(34,36,40,46)(35,45,37,48)(41,44,43,47)

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=hi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10154506);