#10460567

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

4620 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)(3,8)(4,6)(5,7)(9,10,14,18)(11,16,12,15)(13,17,22,21)(19,24,20,23)(25,26,31,35)(27,33,34,30)(28,37,29,32)(36,38,39,40);(1,3,5,4)(2,6,7,8)(11,12)(15,16)(19,20)(23,24)(25,27)(26,30)(28,36)(29,39)(31,34)(32,38)(33,35)(37,40);(9,11,14,12)(10,15,18,16)(13,19,22,20)(17,23,21,24)(25,28,31,29)(26,32,35,37)(27,36,34,39)(30,38,33,40);(1,2)(3,8)(4,6)(5,7)(9,12,14,11)(10,16,18,15)(13,19,22,20)(17,23,21,24)(25,29,31,28)(26,32,35,37)(27,36,34,39)(30,40,33,38);(1,2)(3,6)(4,8)(5,7)(9,13)(10,17)(11,19)(12,20)(14,22)(15,23)(16,24)(18,21)(25,27)(26,33)(28,36)(29,39)(30,35)(31,34)(32,40)(37,38);(1,4,5,3)(2,6,7,8)(9,10)(11,15)(12,16)(13,21)(14,18)(17,22)(19,24)(20,23)(25,30,31,33)(26,34,35,27)(28,38,29,40)(32,39,37,36)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10367763);