#8445845

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	#36791	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4391 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=g,d^2=gh,e^2=i,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, 8353041);