#8463461

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	#2703	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=g,e^2=i,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8370657);