#8720416

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	#38816	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=ghi,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8627612);