#8322234

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	#59308	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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)(3,4,11,7)(5,8,13,15)(10,12,16,14)(17,18,23,22)(19,26,21,24)(20,28,30,25)(27,31,29,32);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,19,23,21)(18,24,22,26)(20,29,30,27)(25,32,28,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,45)(46,48);(5,13)(8,15)(10,16)(12,14)(18,22)(24,26)(25,28)(31,32)(34,39)(37,47)(40,45)(41,48);(1,4,6,7)(2,3,9,11)(5,12,13,14)(8,10,15,16)(17,20,23,30)(18,25,22,28)(19,27,21,29)(24,31,26,32)(33,35)(34,40)(36,42)(37,41)(38,44)(39,45)(43,46)(47,48);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,16)(14,15)(17,21,23,19)(18,26,22,24)(20,27,30,29)(25,31,28,32)(33,36)(34,41)(35,43)(37,45)(38,46)(39,48)(40,47)(42,44);(1,5,6,13)(2,8,9,15)(3,10,11,16)(4,12,7,14)(17,22,23,18)(19,26,21,24)(20,28,30,25)(27,32,29,31)(33,37,38,47)(34,42,39,43)(35,41,44,48)(36,45,46,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=g,e^2=h,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, 8229430);