#10481010

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

4176 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=gi,f^2=g,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, 10388206);