#8818577

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

4223 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,d^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8725773);