#8438117

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
prime radical	#39120	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4630 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)(11,20)(12,17)(13,23)(14,24)(15,19)(16,22)(18,21)(28,36)(32,39)(34,40)(37,38);(3,7)(6,8)(9,11)(10,16)(12,14)(13,21)(15,22)(17,18)(19,20)(23,24)(25,26)(27,31)(28,32)(29,33)(30,35)(34,38)(36,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,13)(10,17,19,23)(11,14,22,21)(16,18,20,24)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,3,4,7)(2,6,5,8)(9,13,15,12)(10,17,19,23)(11,21,22,14)(16,18,20,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(9,14,15,21)(10,18,19,24)(11,12,22,13)(16,17,20,23)(25,28,29,36)(26,32,33,39)(27,34,35,40)(30,37,31,38);(9,10)(11,16)(12,17)(13,23)(14,18)(15,19)(20,22)(21,24)(25,26)(27,30)(28,32)(29,33)(31,35)(34,37)(36,39)(38,40)

Polycyclic presentation:

2^9|c^2=g,d^2=gh,e^2=gi,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, 8345313);