#7514233

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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

3208 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421429);