#8806904

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

4192 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)(3,5)(4,9)(6,7)(8,10)(11,13)(12,14)(15,16)(17,18)(19,22)(20,27)(21,23)(24,29)(25,31)(26,32)(28,30)(33,34,38,42)(35,36,40,39)(37,41,46,48)(43,44,47,45);(2,7)(4,12)(5,8)(11,16)(17,19,21,25)(18,22,23,31)(20,28,29,26)(24,32,27,30)(33,35,38,40)(34,39,42,36)(37,43,46,47)(41,45,48,44);(2,7)(3,10)(4,12)(13,15)(18,23)(19,25)(24,27)(26,28)(33,36,38,39)(34,40,42,35)(37,45,46,44)(41,43,48,47);(1,3,6,10)(2,8,7,5)(4,11,12,16)(9,15,14,13)(17,18)(19,22)(20,24)(21,23)(25,31)(26,30)(27,29)(28,32)(33,36)(34,35)(37,44)(38,39)(40,42)(41,43)(45,46)(47,48);(1,4,6,12)(2,9,7,14)(3,11,10,16)(5,13,8,15)(17,19)(18,22)(20,26)(21,25)(23,31)(24,30)(27,32)(28,29)(33,37)(34,41)(35,43)(36,44)(38,46)(39,45)(40,47)(42,48);(1,5,6,8)(2,10,7,3)(4,13,12,15)(9,16,14,11)(17,20,21,29)(18,24,23,27)(19,26,25,28)(22,30,31,32)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=ghi,d^2=hi,e^2=hi,f^2=h,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, 8714100);