#7710089

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	#36424	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4410 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=i,e^2=gh,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7617285);