#8324466

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4210 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)(3,11)(4,14)(5,10)(6,7)(8,12)(9,15)(13,16)(18,19)(23,25)(24,26)(30,31)(33,34,39,41)(35,40,43,37)(36,42,46,48)(38,44,45,47);(2,7)(3,12)(9,14)(10,13)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,35,39,43)(34,40,41,37)(36,45,46,38)(42,47,48,44);(2,7)(5,16)(8,11)(9,14)(21,29)(24,26)(27,32)(30,31)(34,41)(35,43)(38,45)(42,48);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,36)(34,42)(35,38)(37,47)(39,46)(40,44)(41,48)(43,45);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,37,39,40)(34,43,41,35)(36,44,46,47)(38,48,45,42);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,38)(34,44)(35,36)(37,48)(39,45)(40,42)(41,47)(43,46)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=g,e^2=h,f^2=hi,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, 8231662);