#8932726

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

4133 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(19,25)(20,28)(23,26)(24,29)(34,36)(37,45)(39,41)(42,48);(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,18)(19,26)(20,29)(21,22)(23,25)(24,28)(27,31)(30,32)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(4,12)(7,14)(10,16)(13,15)(18,22)(23,26)(24,29)(31,32)(33,34,38,36)(35,39,43,41)(37,44,45,40)(42,47,48,46);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,23)(20,30)(21,25)(22,26)(24,32)(27,28)(29,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,31)(25,30)(26,32)(33,36,38,34)(35,39,43,41)(37,44,45,40)(42,46,48,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|b^2=g,c^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839922);