#7715473

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	#38903	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4197 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7622669);