#10247627

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4006 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)(3,8)(4,11)(5,13)(6,7)(9,15)(10,12)(14,16)(17,18)(19,21)(20,23)(22,28)(24,26)(25,30)(27,31)(29,32)(33,34,39,38)(35,43,45,40)(36,41,47,46)(37,48,44,42);(1,3,6,11)(2,4,7,8)(5,10,15,14)(9,12,13,16)(17,19)(18,24)(20,21)(22,27)(23,26)(25,31)(28,29)(30,32)(33,35)(34,40)(36,44)(37,47)(38,43)(39,45)(41,48)(42,46);(2,7)(4,8)(9,13)(12,16)(18,20)(19,26)(25,28)(27,32)(33,36,39,47)(34,41,38,46)(35,44,45,37)(40,48,43,42);(1,4)(2,3)(5,12)(6,8)(7,11)(9,10)(13,14)(15,16)(17,20,23,18)(19,21,26,24)(22,28,30,25)(27,29,32,31)(33,37)(34,42)(35,36)(38,48)(39,44)(40,41)(43,46)(45,47);(1,3)(2,8)(4,7)(5,14)(6,11)(9,12)(10,15)(13,16)(17,21)(18,19)(20,26)(22,29)(23,24)(25,27)(28,32)(30,31)(33,35)(34,43)(36,44)(37,47)(38,40)(39,45)(41,42)(46,48);(1,5)(2,9)(3,10)(4,12)(6,15)(7,13)(8,16)(11,14)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38,39,34)(35,43,45,40)(36,46,47,41)(37,48,44,42)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=i,f^2=gh,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, 10154823);