#8925286

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

4166 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)(4,10)(5,15)(6,8)(7,13)(11,12)(14,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32)(35,41)(39,46)(42,44)(47,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,9)(10,16)(14,15)(17,18,21,24)(19,22,27,25)(20,23,28,30)(26,29,32,31)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(2,8)(5,10)(7,9)(12,16)(20,28)(23,30)(26,32)(29,31)(34,38)(39,46)(40,43)(47,48);(1,3)(2,9)(4,14)(5,12)(6,13)(7,8)(10,16)(11,15)(17,19,21,27)(18,22,24,25)(20,26,28,32)(23,29,30,31)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,4)(2,10)(3,11)(5,8)(6,15)(7,16)(9,12)(13,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32)(33,36)(34,40)(35,42)(37,45)(38,43)(39,47)(41,44)(46,48);(1,5,6,10)(2,4,8,15)(3,12,13,16)(7,11,9,14)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,45)(44,48)

Polycyclic presentation:

2^9|b^2=gh,d^2=gh,f^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, 8832482);