#10106128

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

4255 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,28)(20,21)(22,32)(23,25)(24,29)(26,31)(27,30)(33,34,38,41)(35,37,43,39)(36,40,46,48)(42,47,44,45);(3,9)(7,13)(10,15)(14,16)(17,19,23,29)(18,24,25,28)(20,30,31,22)(21,32,26,27)(33,35)(34,37)(36,44)(38,43)(39,41)(40,45)(42,46)(47,48);(2,6)(7,13)(8,11)(14,16)(18,25)(19,29)(20,31)(27,32)(33,35,38,43)(34,39,41,37)(36,42,46,44)(40,45,48,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20)(18,26)(19,30)(21,25)(22,29)(23,31)(24,27)(28,32)(33,35)(34,37)(36,44)(38,43)(39,41)(40,45)(42,46)(47,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,21,23,26)(18,20,25,31)(19,27,29,32)(22,28,30,24)(33,36)(34,40)(35,42)(37,47)(38,46)(39,45)(41,48)(43,44);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22,23,30)(18,27,25,32)(19,20,29,31)(21,28,26,24)(33,37)(34,35)(36,45)(38,39)(40,44)(41,43)(42,48)(46,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gi,e^2=g,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, 10013324);