#8838380

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3820 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)(3,7)(4,8)(5,14)(6,10)(9,15)(11,12)(13,16)(17,18,23,20)(19,26,24,21)(22,28,30,25)(27,29,31,32)(34,39)(35,43)(36,46)(42,47);(1,3)(2,7)(4,5)(6,11)(8,14)(9,16)(10,12)(13,15)(19,24)(21,26)(22,30)(25,28)(33,34,38,39)(35,44,43,40)(36,41,46,48)(37,47,45,42);(3,11)(4,13)(7,12)(8,16)(17,19,23,24)(18,21,20,26)(22,27,30,31)(25,29,28,32)(33,35,38,43)(34,40,39,44)(36,45,46,37)(41,47,48,42);(1,4,6,13)(2,8,10,16)(3,5,11,15)(7,14,12,9)(17,20,23,18)(19,21,24,26)(22,28,30,25)(27,29,31,32)(33,36)(34,41)(35,45)(37,43)(38,46)(39,48)(40,47)(42,44);(1,5)(2,9)(3,4)(6,15)(7,16)(8,12)(10,14)(11,13)(17,21)(18,24)(19,20)(22,29)(23,26)(25,31)(27,28)(30,32)(33,37,38,45)(34,42,39,47)(35,36,43,46)(40,41,44,48);(1,2)(3,12)(4,8)(5,9)(6,10)(7,11)(13,16)(14,15)(17,22,23,30)(18,25,20,28)(19,27,24,31)(21,29,26,32)(33,36)(34,41)(35,45)(37,43)(38,46)(39,48)(40,47)(42,44)

Polycyclic presentation:

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

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, 8745576);