#8824122

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

3914 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=h,d^2=hi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8731318);