#7514033

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

3148 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)(11,21,13,27)(12,17,24,32)(14,29,30,19)(16,28,18,23)(22,35,25,38)(26,40,37,34)(31,39,33,36);(1,2)(3,4)(5,6)(7,8)(9,11,15,13)(10,16,20,18)(12,22,24,25)(14,28,30,23)(17,31,32,33)(19,27,29,21)(26,39,37,36)(34,38,40,35);(6,8)(9,12)(10,17)(11,22)(13,25)(14,26)(15,24)(16,31)(18,33)(19,34)(20,32)(21,35)(23,36)(27,38)(28,39)(29,40)(30,37);(2,4)(6,8)(12,24)(17,32)(22,25)(26,37)(31,33)(34,40)(35,38)(36,39);(9,13,15,11)(10,18,20,16)(12,25,24,22)(14,23,30,28)(17,33,32,31)(19,21,29,27)(26,36,37,39)(34,35,40,38)

Polycyclic presentation:

2^9|a^2=i,b^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=cg,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, 7421229);