#8931371

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

4123 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)(3,8)(4,9)(5,10)(6,7)(11,16)(12,15)(13,14)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32)(35,41)(39,46)(42,44)(47,48);(2,7)(4,14)(8,16)(10,12)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(3,11)(5,15)(8,16)(10,12)(18,20)(22,30)(24,26)(27,29)(34,38)(35,41)(40,43)(42,44);(1,3,6,11)(2,8,7,16)(4,10,14,12)(5,13,15,9)(17,20,23,18)(19,26,21,24)(22,28,30,25)(27,32,29,31)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,4,6,14)(2,9,7,13)(3,12,11,10)(5,8,15,16)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,36,37,45)(34,40,38,43)(35,42,41,44)(39,47,46,48);(1,5,6,15)(2,10,7,12)(3,13,11,9)(4,8,14,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,45)(44,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8838567);