#8182111

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4413 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,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, 8089307);