#7514313

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

Polycyclic presentation:

2^9|a^2=i,c^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=cgi,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, 7421509);