#7648484

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	#37620	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4318 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)(13,21)(16,23)(19,22)(20,24)(25,26)(27,32)(28,34)(29,38)(30,33)(31,35)(36,40)(37,39);(1,3,5,8)(2,6,4,7)(9,10)(11,18)(12,15)(13,16)(14,17)(19,24)(20,22)(21,23)(25,27,31,29)(26,32,35,38)(28,36,30,37)(33,39,34,40);(9,11,14,15)(10,12,17,18)(13,19,21,22)(16,20,23,24)(25,28,31,30)(26,33,35,34)(27,36,29,37)(32,39,38,40);(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,23)(22,24)(25,29,31,27)(26,32,35,38)(28,37,30,36)(33,39,34,40);(1,4,5,2)(3,7,8,6)(9,12)(10,15)(11,17)(13,20)(14,18)(16,22)(19,23)(21,24)(25,26)(27,32)(28,33)(29,38)(30,34)(31,35)(36,39)(37,40);(9,13,14,21)(10,16,17,23)(11,19,15,22)(12,20,18,24)(25,30,31,28)(26,34,35,33)(27,37,29,36)(32,40,38,39)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=hi,d^2=h,e^2=g,f^2=hi,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, 7555680);