#7518975

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2631 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=i,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^c=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, 7426171);