#8436981

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4288 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=hi,d^2=g,e^2=h,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, 8344177);