#8127381

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3878 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=gi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8034577);