#7514115

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#35757	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3138 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)(9,10,15,20)(11,17,23,33)(12,18,14,16)(13,26,27,30)(19,25,34,28)(21,29,24,32)(22,36,38,39)(31,37,40,35);(1,2)(3,4)(5,6)(7,8)(10,16)(12,14)(17,29)(18,20)(19,30)(21,24)(25,28)(26,34)(31,39)(32,33)(35,37)(36,40);(6,8)(9,11)(10,17)(12,24)(13,22)(14,21)(15,23)(16,29)(18,32)(19,31)(20,33)(25,37)(26,36)(27,38)(28,35)(30,39)(34,40);(2,4)(6,8)(9,12,15,14)(10,18,20,16)(11,21,23,24)(13,25,27,28)(17,29,33,32)(19,26,34,30)(22,35,38,37)(31,39,40,36);(9,13,15,27)(10,19,20,34)(11,22,23,38)(12,25,14,28)(16,30,18,26)(17,31,33,40)(21,35,24,37)(29,39,32,36)

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,f^a=fi,c^b=cg,d^b=dh,f^b=fi,d^c=di,f^e=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421311);