#8932521

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

4470 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=gi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839717);