#8928620

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

4192 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)(11,14)(12,21)(13,17)(15,18)(16,20)(19,24)(22,23)(25,26,31,28)(27,34,30,32)(29,33,37,36)(35,40,38,39);(2,3)(6,7)(9,11)(10,14)(12,13)(15,19)(16,22)(17,21)(18,24)(20,23)(29,37)(33,36)(35,38)(39,40);(1,3,5,2)(4,7,8,6)(11,19)(12,20)(14,24)(16,21)(25,27,31,30)(26,32,28,34)(29,35,37,38)(33,39,36,40);(1,4,5,8)(2,6,3,7)(9,12)(10,16)(11,13)(14,22)(15,20)(17,24)(18,21)(19,23)(25,28,31,26)(27,34,30,32)(29,36,37,33)(35,40,38,39);(9,13)(10,17)(11,20)(12,19)(14,16)(15,23)(18,22)(21,24)(25,29,31,37)(26,33,28,36)(27,35,30,38)(32,39,34,40);(9,14)(10,11)(12,22)(13,16)(15,24)(17,20)(18,19)(21,23)(25,30,31,27)(26,34,28,32)(29,38,37,35)(33,40,36,39)

Polycyclic presentation:

2^9|a^2=gh,c^2=gh,d^2=gh,e^2=h,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, 8835816);