#8323311

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	#52249	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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)(3,10,4,7)(5,8,13,15)(11,16,12,14)(17,18,23,26)(19,22,25,31)(20,21,24,30)(27,28,29,32)(33,34,39,44)(35,42,37,40)(36,45,47,41)(38,43,48,46);(1,3,6,4)(2,7,9,10)(5,11,13,12)(8,14,15,16)(17,19)(18,22)(20,27)(21,28)(23,25)(24,29)(26,31)(30,32)(33,35,39,37)(34,40,44,42)(36,46,47,43)(38,45,48,41);(5,13)(8,15)(11,12)(14,16)(19,25)(22,31)(27,29)(28,32)(33,35,39,37)(34,40,44,42)(36,43,47,46)(38,41,48,45);(1,2)(3,7)(4,10)(5,8)(6,9)(11,14)(12,16)(13,15)(17,20)(18,21)(19,27)(22,28)(23,24)(25,29)(26,30)(31,32)(33,36)(34,41)(35,43)(37,46)(38,42)(39,47)(40,48)(44,45);(1,4,6,3)(2,7,9,10)(5,12,13,11)(8,14,15,16)(17,21)(18,24)(19,28)(20,26)(22,29)(23,30)(25,32)(27,31)(33,37,39,35)(34,42,44,40)(36,43,47,46)(38,41,48,45);(1,5,6,13)(2,8,9,15)(3,11,4,12)(7,14,10,16)(17,22,23,31)(18,25,26,19)(20,28,24,32)(21,29,30,27)(33,38,39,48)(34,43,44,46)(35,45,37,41)(36,42,47,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8230507);