#9015382

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

4229 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

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, 8922578);