#8924669

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

4189 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)(3,4)(5,13)(6,7)(8,10)(9,14)(11,15)(12,16)(17,18,23,25)(19,24,29,32)(20,26,30,31)(21,27,28,22)(33,34,39,38)(35,44,47,40)(36,41,46,42)(37,43,45,48);(2,7)(4,8)(9,13)(12,15)(17,19,23,29)(18,24,25,32)(20,28,30,21)(22,31,27,26)(33,35)(34,40)(36,45)(37,46)(38,44)(39,47)(41,43)(42,48);(2,7)(3,10)(5,14)(12,15)(18,25)(19,29)(20,30)(22,27)(33,36,39,46)(34,41,38,42)(35,45,47,37)(40,43,44,48);(1,3,6,10)(2,8,7,4)(5,11,14,16)(9,15,13,12)(17,20)(18,26)(19,28)(21,29)(22,24)(23,30)(25,31)(27,32)(33,36)(34,42)(35,45)(37,47)(38,41)(39,46)(40,48)(43,44);(1,4,6,8)(2,3,7,10)(5,12,14,15)(9,11,13,16)(17,21,23,28)(18,27,25,22)(19,30,29,20)(24,31,32,26)(33,37,39,45)(34,43,38,48)(35,46,47,36)(40,41,44,42);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,22,23,27)(18,28,25,21)(19,31,29,26)(20,24,30,32)(33,38,39,34)(35,44,47,40)(36,41,46,42)(37,48,45,43)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=ghi,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,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, 8831865);