#8109990

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4374 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:

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

Polycyclic presentation:

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

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, 8017186);