#7732420

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4200 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=h,e^2=hi,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, 7639616);