#8436630

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,22,26)(19,23,21,25)(20,28,29,24)(27,32,30,31)(33,34,38,41)(35,36,43,39)(37,40,45,47)(42,44,48,46);(2,3)(7,9)(8,10)(14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,29)(30,32)(33,35,38,43)(34,39,41,36)(37,42,45,48)(40,46,47,44);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19,22,21)(18,23,26,25)(20,27,29,30)(24,31,28,32)(33,35)(34,36)(37,42)(38,43)(39,41)(40,44)(45,48)(46,47);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,24)(19,27)(21,30)(22,29)(23,31)(25,32)(26,28)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,46)(47,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,31)(28,29)(30,32);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21,22,19)(18,25,26,23)(20,27,29,30)(24,31,28,32)(33,37,38,45)(34,40,41,47)(35,42,43,48)(36,44,39,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=h,f^2=g,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, 8343826);