#7614082

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45267	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2617 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,d^2=h,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521278);