#8458104

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

4646 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=gi,d^2=gh,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8365300);