#8012059

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

4612 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)(9,10,15,13)(11,19,20,16)(12,17,22,23)(14,18,21,24)(25,26)(27,33)(28,34)(29,36)(30,31)(32,37)(35,39)(38,40);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,21)(13,19)(14,22)(15,20)(17,18)(23,24)(26,31)(28,38)(32,33)(35,36);(11,20)(14,21)(16,19)(18,24)(25,27,30,37)(26,32,31,33)(28,36,38,35)(29,34,39,40);(1,2,5,4)(3,6,8,7)(9,12,15,22)(10,17,13,23)(11,21,20,14)(16,18,19,24)(25,27)(26,33)(28,36)(29,34)(30,37)(31,32)(35,38)(39,40);(9,13,15,10)(11,19,20,16)(12,17,22,23)(14,24,21,18)(25,28,30,38)(26,34,31,40)(27,36,37,35)(29,32,39,33);(1,4,5,2)(3,7,8,6)(9,14,15,21)(10,18,13,24)(11,22,20,12)(16,17,19,23)(25,29)(26,35)(27,34)(28,32)(30,39)(31,36)(33,38)(37,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=gh,e^2=hi,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, 7919255);