#10247529

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	#56317	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4004 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)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,18)(19,25)(20,22)(21,24)(23,27)(26,31)(28,29)(30,32);(3,9)(6,13)(10,15)(14,16)(17,19,23,21)(18,24,27,25)(20,30,31,28)(22,29,26,32)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,45,47);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,19,23,21)(18,25,27,24)(20,28,31,30)(22,29,26,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20)(18,26)(19,28)(21,30)(22,27)(23,31)(24,29)(25,32)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,21,23,19)(18,24,27,25)(20,28,31,30)(22,29,26,32)(33,35)(34,36)(37,45)(38,43)(39,41)(40,47)(42,46)(44,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,22,23,26)(18,20,27,31)(19,29,21,32)(24,30,25,28)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=i,e^2=g,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, 10154725);