#10166776

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

4283 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,15)(14,16)(18,19)(23,25)(24,26)(31,32)(33,34)(35,37)(36,39)(38,44)(40,42)(41,46)(43,47)(45,48);(2,7)(4,14)(8,11)(10,13)(17,18,22,19)(20,23,28,25)(21,24,30,26)(27,31,29,32)(33,35)(34,40)(36,37)(38,43)(39,42)(41,47)(44,45)(46,48);(2,7)(3,12)(9,16)(10,13)(34,36)(35,42)(41,44)(43,48);(1,3)(2,8)(4,10)(5,9)(6,12)(7,11)(13,14)(15,16)(17,19,22,18)(20,23,28,25)(21,26,30,24)(27,31,29,32)(33,36,39,34)(35,37,42,40)(38,44,46,41)(43,45,48,47);(1,4,6,14)(2,9,7,16)(3,13,12,10)(5,8,15,11)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,37)(34,35)(36,42)(38,45)(39,40)(41,43)(44,48)(46,47);(1,5,6,15)(2,10,7,13)(3,9,12,16)(4,11,14,8)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(33,38,39,46)(34,41,36,44)(35,43,42,48)(37,45,40,47)

Polycyclic presentation:

2^9|b^2=gh,d^2=ghi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10073972);