#7514053

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dgi,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7421249);