#8505427

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×#120	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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=i,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412623);