#7732321

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	#35568	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4290 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)(9,10,16,15)(11,20,14,17)(12,18,13,19)(21,24,22,23)(25,26)(27,28)(29,37)(30,33)(31,32)(34,39)(35,40)(36,38);(1,2,5,6)(3,7,4,8)(9,11,16,14)(10,17,15,20)(12,22,13,21)(18,24,19,23)(26,32)(27,30)(34,37)(35,38);(1,3,5,4)(2,7,6,8)(9,12,16,13)(10,18,15,19)(11,21,14,22)(17,23,20,24)(25,27,31,30)(26,33,32,28)(29,38,39,35)(34,36,37,40);(1,4,5,3)(2,8,6,7)(9,13,16,12)(10,19,15,18)(11,21,14,22)(17,23,20,24)(25,28,31,33)(26,27,32,30)(29,36,39,40)(34,35,37,38);(9,14,16,11)(10,17,15,20)(12,21,13,22)(18,24,19,23)(25,29,31,39)(26,34,32,37)(27,35,30,38)(28,36,33,40);(9,15,16,10)(11,17,14,20)(12,19,13,18)(21,23,22,24)(25,30,31,27)(26,33,32,28)(29,38,39,35)(34,40,37,36)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7639517);