#8001028

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

4339 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7908224);