#8182062

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	#36147	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4534 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=g,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8089258);