#8824406

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
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3866 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=hi,e^2=ghi,f^2=gi,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, 8731602);