#7917819

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	#39860	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4569 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:

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

Polycyclic presentation:

2^9|a^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7825015);