#8000990

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

4447 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=gh,e^2=gh,f^2=h,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, 7908186);