#8926285

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

4122 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)(3,11)(4,8)(5,6)(7,9)(10,16)(12,15)(13,14)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,31)(30,32)(35,41)(40,47)(42,45)(43,48);(1,3,7,4)(2,8,9,11)(5,12,10,14)(6,13,16,15)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28)(33,34,38,39)(35,40,41,47)(36,44,46,37)(42,43,45,48);(2,9)(6,16)(8,11)(13,15)(21,29)(24,31)(26,32)(28,30)(34,39)(37,44)(40,47)(43,48);(1,4,7,3)(2,8,9,11)(5,12,10,14)(6,15,16,13)(17,18)(19,23)(20,27)(21,24)(22,25)(26,30)(28,32)(29,31)(33,35)(34,40)(36,45)(37,43)(38,41)(39,47)(42,46)(44,48);(1,5,7,10)(2,6,9,16)(3,12,4,14)(8,13,11,15)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,36)(34,37)(35,42)(38,46)(39,44)(40,48)(41,45)(43,47);(1,6)(2,10)(3,13)(4,15)(5,9)(7,16)(8,14)(11,12)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,37)(34,36)(35,43)(38,44)(39,46)(40,45)(41,48)(42,47)

Polycyclic presentation:

2^9|b^2=gi,d^2=h,e^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, 8833481);