#10247629

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	#53128	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4008 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)(3,9)(4,8)(5,10)(6,7)(11,14)(12,16)(13,15)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32)(33,34,39,38)(35,43,45,40)(36,41,44,46)(37,42,47,48);(2,7)(3,10)(4,12)(14,15)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,35,39,45)(34,40,38,43)(36,37,44,47)(41,48,46,42);(2,7)(3,10)(8,16)(11,13)(18,19)(23,25)(24,26)(30,31)(33,36,39,44)(34,41,38,46)(35,37,45,47)(40,48,43,42);(1,3)(2,5)(4,13)(6,10)(7,9)(8,14)(11,12)(15,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,35)(34,40)(36,37)(38,43)(39,45)(41,48)(42,46)(44,47);(1,4)(2,8)(3,11)(5,15)(6,12)(7,16)(9,14)(10,13)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,37)(34,42)(35,44)(36,45)(38,48)(39,47)(40,41)(43,46);(1,5)(2,3)(4,14)(6,9)(7,10)(8,13)(11,16)(12,15)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,38,39,34)(35,43,45,40)(36,46,44,41)(37,42,47,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,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, 10154825);