#8152163

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	#56350	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4353 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)(11,19)(12,22)(13,24)(14,20)(15,18)(16,21)(17,23)(25,26,30,34)(27,35,38,31)(28,39,37,32)(29,33,36,40);(3,7)(6,8)(9,11)(10,16)(12,20)(13,23)(14,17)(15,21)(18,19)(22,24)(25,27,30,38)(26,31,34,35)(28,29,37,36)(32,33,39,40);(1,2)(3,6)(4,5)(7,8)(9,12,15,23)(10,17,18,22)(11,20,21,13)(14,19,24,16)(25,28,30,37)(26,32,34,39)(27,36,38,29)(31,40,35,33);(1,3)(2,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,20)(16,22)(17,19)(18,24)(21,23)(25,27)(26,31)(28,36)(29,37)(30,38)(32,40)(33,39)(34,35);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,20)(13,23)(14,17)(15,21)(18,19)(22,24)(25,26)(27,31)(28,32)(29,33)(30,34)(35,38)(36,40)(37,39);(9,14)(10,13)(11,17)(12,19)(15,24)(16,23)(18,20)(21,22)(25,29,30,36)(26,33,34,40)(27,37,38,28)(31,39,35,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8059359);