#7514112

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3139 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)(11,20)(12,17)(13,26)(14,28)(15,19)(16,22)(18,29)(21,34)(23,32)(24,37)(25,39)(27,31)(30,35)(33,36)(38,40);(1,2)(3,4)(5,6)(7,8)(9,11,15,13)(10,16,19,31)(12,21,23,24)(14,27,29,22)(17,30,32,40)(18,26,28,20)(25,38,36,35)(33,37,39,34);(6,8)(9,12)(10,17)(11,21)(13,24)(14,25)(15,23)(16,30)(18,33)(19,32)(20,34)(22,35)(26,37)(27,38)(28,39)(29,36)(31,40);(2,4)(6,8)(12,23)(17,32)(21,24)(25,36)(30,40)(33,39)(34,37)(35,38);(9,13,15,11)(10,16,19,31)(12,24,23,21)(14,22,29,27)(17,30,32,40)(18,26,28,20)(25,35,36,38)(33,37,39,34)

Polycyclic presentation:

2^9|b^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, 7421308);