#7648341

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=i,d^2=g,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

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, 7555537);