#8889061

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	#46338	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4191 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)(3,10,8,4)(5,13,14,9)(11,12,15,16)(19,25)(22,26)(27,31)(30,32)(33,34,39,36)(35,41,37,40)(38,44,46,42)(43,47,45,48);(2,7)(4,10)(5,14)(11,15)(17,18)(19,22)(20,28)(21,24)(23,29)(25,26)(27,32)(30,31)(33,34,39,36)(35,41,37,40)(38,44,46,42)(43,47,45,48);(1,3,6,8)(2,4,7,10)(5,11,14,15)(9,12,13,16)(17,18,21,24)(19,22,25,26)(20,23,29,28)(27,30,31,32)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,29)(30,31)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,19,21,25)(18,22,24,26)(20,27,29,31)(23,30,28,32)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,20,21,29)(18,23,24,28)(19,27,25,31)(22,30,26,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=gh,e^2=i,f^2=ghi,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, 8796257);