#8688330

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

4445 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)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(25,26)(27,33)(28,32)(29,34)(30,35)(31,38)(36,37)(39,40);(2,6)(3,8)(10,15)(12,16)(17,24)(20,21)(25,27)(26,31)(28,36)(29,37)(30,38)(32,39)(33,35)(34,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,16)(12,15)(13,18)(14,19)(17,21)(20,24)(22,23)(25,28,30,34)(26,32,35,29)(27,36,38,40)(31,39,33,37);(1,3)(2,4)(5,8)(6,7)(9,12,14,16)(10,11,15,19)(13,21,23,20)(17,22,24,18)(25,27)(26,33)(28,36)(29,39)(30,38)(31,35)(32,37)(34,40);(1,4)(2,3)(5,7)(6,8)(9,11)(10,16)(12,15)(13,22)(14,19)(17,20)(18,23)(21,24)(25,27)(26,33)(28,36)(29,39)(30,38)(31,35)(32,37)(34,40);(9,13,14,23)(10,17,15,24)(11,18,19,22)(12,20,16,21)(25,29,30,32)(26,34,35,28)(27,37,38,39)(31,40,33,36)

Polycyclic presentation:

2^9|c^2=gh,d^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8595526);