#8932722

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

4141 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)(4,8)(5,6)(7,11)(10,16)(12,15)(13,14)(19,25)(20,28)(23,26)(24,29)(34,40)(37,43)(41,47)(45,48);(1,3)(2,7)(4,12)(5,9)(6,11)(8,16)(10,14)(13,15)(17,18)(19,26)(20,29)(21,22)(23,25)(24,28)(27,31)(30,32)(33,34)(35,37)(36,43)(38,41)(39,40)(42,45)(44,48)(46,47);(2,5)(7,9)(8,13)(15,16)(18,22)(23,26)(24,29)(31,32)(33,35,39,36)(34,37,40,43)(38,44,46,42)(41,48,47,45);(1,4,6,14)(2,8,5,13)(3,10,11,12)(7,15,9,16)(17,19)(18,23)(20,30)(21,25)(22,26)(24,32)(27,28)(29,31)(33,36,39,35)(34,37,40,43)(38,44,46,42)(41,45,47,48);(1,3)(2,7)(4,12)(5,9)(6,11)(8,16)(10,14)(13,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,31)(25,30)(26,32)(33,37)(34,36)(35,40)(38,45)(39,43)(41,44)(42,47)(46,48);(1,5,6,2)(3,9,11,7)(4,13,14,8)(10,16,12,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,38,39,46)(34,41,40,47)(35,42,36,44)(37,45,43,48)

Polycyclic presentation:

2^9|c^2=gh,d^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, 8839918);