#8821853

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

4068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=gh,e^2=ghi,f^2=hi,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, 8729049);