#8755426

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

4153 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=hi,d^2=hi,e^2=h,f^2=g,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, 8662622);