#8417229

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3847 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324425);