#8436836

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×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8344032);