#8932277

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

4364 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=h,e^2=gi,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, 8839473);