#8180472

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4001 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8087668);