#8745965

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	#52656	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3946 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8653161);