#8824460

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4001 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)(11,16)(12,13)(14,15)(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32)(33,34,37,40)(35,41,43,38)(36,39,45,48)(42,47,44,46);(1,3,7,4)(2,8,9,10)(5,14,11,12)(6,15,16,13)(17,19,23,21)(18,24,20,25)(22,29,30,27)(26,32,28,31)(33,35,37,43)(34,38,40,41)(36,44,45,42)(39,47,48,46);(2,9)(3,4)(5,11)(13,15)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,45)(47,48);(1,4,7,3)(2,10,9,8)(5,14,11,12)(6,15,16,13)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48);(1,5,7,11)(2,6,9,16)(3,12,4,14)(8,13,10,15)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31)(33,36,37,45)(34,39,40,48)(35,42,43,44)(38,46,41,47);(1,6)(2,11)(3,13)(4,15)(5,9)(7,16)(8,14)(10,12)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,d^2=gi,e^2=i,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, 8731656);