#8456013

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

4695 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)(6,8)(9,10,16,19)(11,17,14,15)(12,22,13,18)(20,23,21,24)(25,26)(27,35)(28,32)(29,37)(30,33)(31,34)(36,39)(38,40);(3,7)(6,8)(9,11,16,14)(10,17,19,15)(12,21,13,20)(18,23,22,24)(25,27)(26,29)(28,36)(30,32)(31,37)(33,38)(34,35)(39,40);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,18,19,22)(11,20,14,21)(15,23,17,24)(25,28,31,38)(26,32,34,40)(27,36,37,33)(29,30,35,39);(1,3,4,7)(2,6,5,8)(9,13,16,12)(10,18,19,22)(11,21,14,20)(15,23,17,24)(25,29)(26,27)(28,30)(31,35)(32,36)(33,40)(34,37)(38,39);(1,2)(3,6)(4,5)(7,8)(9,14,16,11)(10,15,19,17)(12,21,13,20)(18,23,22,24)(25,27)(26,29)(28,36)(30,32)(31,37)(33,38)(34,35)(39,40);(9,15)(10,11)(12,23)(13,24)(14,19)(16,17)(18,20)(21,22)(25,30)(26,33)(27,32)(28,35)(29,38)(31,39)(34,36)(37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=h,e^2=i,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, 8363209);