#8323536

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4157 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)(3,8)(4,10)(5,6)(7,9)(11,13)(12,15)(14,16)(17,18,23,26)(19,22,25,31)(20,21,24,30)(27,28,29,32)(33,34,40,44)(35,45,48,41)(36,42,37,39)(38,43,47,46);(1,3,7,6)(2,8,9,5)(4,12,14,11)(10,15,16,13)(17,19)(18,22)(20,27)(21,28)(23,25)(24,29)(26,31)(30,32)(33,35)(34,41)(36,43)(37,46)(38,42)(39,47)(40,48)(44,45);(2,9)(3,6)(10,16)(11,12)(19,25)(22,31)(27,29)(28,32)(33,36,40,37)(34,42,44,39)(35,43,48,46)(38,45,47,41);(1,4,7,14)(2,10,9,16)(3,11,6,12)(5,15,8,13)(17,20)(18,21)(19,27)(22,28)(23,24)(25,29)(26,30)(31,32)(33,37,40,36)(34,42,44,39)(35,46,48,43)(38,45,47,41);(1,5)(2,3)(4,13)(6,9)(7,8)(10,12)(11,16)(14,15)(17,21)(18,24)(19,28)(20,26)(22,29)(23,30)(25,32)(27,31)(33,38,40,47)(34,43,44,46)(35,42,48,39)(36,45,37,41);(1,6,7,3)(2,5,9,8)(4,12,14,11)(10,15,16,13)(17,22,23,31)(18,25,26,19)(20,28,24,32)(21,29,30,27)(33,39,40,42)(34,36,44,37)(35,47,48,38)(41,43,45,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=gh,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, 8230732);