#7271708

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234485
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42505	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1366 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=g,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7178904);