#7710444

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	#45768	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4371 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)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(28,35)(32,37)(33,39)(38,40);(1,2,5,6)(3,7,4,8)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40);(10,11)(15,17)(16,18)(22,23)(26,30)(31,36)(32,37)(38,40);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,27,29,34)(26,31,30,36)(28,33,35,39)(32,38,37,40);(1,3,5,4)(2,7,6,8)(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,26)(27,31)(28,32)(29,30)(33,38)(34,36)(35,37)(39,40);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,28)(26,32)(27,33)(29,35)(30,37)(31,38)(34,39)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=ghi,e^2=gh,f^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, 7617640);