#9545755

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

4003 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,21,24)(19,22,25,26)(20,23,28,30)(27,29,31,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(2,6)(7,9)(8,12)(14,15)(17,18)(19,22)(20,23)(21,24)(25,26)(27,29)(28,30)(31,32)(33,35,39,36)(34,40,38,41)(37,44,45,43)(42,48,46,47);(2,6)(3,10)(4,13)(14,15)(19,25)(20,28)(22,26)(23,30)(34,38)(35,36)(37,45)(47,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,36,39,35)(34,41,38,40)(37,43,45,44)(42,47,46,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,19)(18,22)(20,27)(21,25)(23,29)(24,26)(28,31)(30,32)(33,37,39,45)(34,42,38,46)(35,43,36,44)(40,47,41,48);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,20)(18,23)(19,27)(21,28)(22,29)(24,30)(25,31)(26,32)(33,38,39,34)(35,41,36,40)(37,46,45,42)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^c=egi,f^c=fgi,e^d=egi

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, 9452951);