#8924961

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

4290 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)(4,5)(6,8)(7,13)(10,15)(11,16)(12,14)(19,25)(23,30)(26,28)(31,32)(33,34)(35,41)(36,42)(37,38)(39,40)(43,46)(44,45)(47,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,9)(10,14)(15,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(34,40)(37,47)(41,42)(43,45);(2,8)(5,15)(7,9)(12,16)(18,22)(23,30)(24,27)(31,32)(33,35,39,36)(34,41,40,42)(37,43,47,45)(38,46,48,44);(1,4,6,10)(2,5,8,15)(3,11,13,14)(7,12,9,16)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,36,39,35)(34,42,40,41)(37,43,47,45)(38,46,48,44);(1,3)(2,9)(4,14)(5,12)(6,13)(7,8)(10,11)(15,16)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,37)(34,38)(35,43)(36,45)(39,47)(40,48)(41,46)(42,44);(1,5)(2,10)(3,12)(4,8)(6,15)(7,14)(9,11)(13,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,38)(34,37)(35,44)(36,46)(39,48)(40,47)(41,45)(42,43)

Polycyclic presentation:

2^9|c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8832157);