#8371251

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	#41495	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=gh,d^2=ghi,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=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, 8278447);