#8505585

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

4368 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)(3,7)(4,8)(5,6)(9,10)(11,17)(12,16)(13,24)(14,18)(15,19)(20,23)(21,22)(25,26,31,28)(27,32,37,35)(29,38,40,33)(30,34,36,39);(2,6)(3,8)(9,11,15,12)(10,16,19,17)(13,22,18,20)(14,21,24,23)(25,27)(26,32)(28,35)(29,30)(31,37)(33,39)(34,38)(36,40);(1,3)(2,4)(5,8)(6,7)(9,11,15,12)(10,17,19,16)(13,20,18,22)(14,21,24,23)(25,26,31,28)(27,35,37,32)(29,33,40,38)(30,34,36,39);(1,3)(2,4)(5,8)(6,7)(9,12,15,11)(10,17,19,16)(13,22,18,20)(14,21,24,23)(25,28,31,26)(27,35,37,32)(29,38,40,33)(30,34,36,39);(1,4)(2,3)(5,7)(6,8)(9,13,15,18)(10,14,19,24)(11,20,12,22)(16,23,17,21)(25,29)(26,33)(27,36)(28,38)(30,37)(31,40)(32,34)(35,39);(9,14)(10,18)(11,21)(12,23)(13,19)(15,24)(16,20)(17,22)(25,30,31,36)(26,34,28,39)(27,29,37,40)(32,38,35,33)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8412781);