#8012076

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

4478 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=gh,e^2=ghi,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, 7919272);