#8445874

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4243 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=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, 8353070);