#8702936

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

4595 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8610132);