#8757385

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

3971 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664581);