#8934847

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

3989 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

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, 8842043);