#8001020

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

4689 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=h,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, 7908216);