#8932183

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

4818 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=g,e^2=gi,f^2=h,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, 8839379);