#7645853

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4221 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gi,d^2=gh,e^2=h,f^2=g,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, 7553049);