#7271693

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234437
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

1358 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)(29,34)(30,35)(31,40)(32,36)(33,38)(37,39);(1,3,6,2)(4,7,5,8)(9,11,13,18)(10,14,16,22)(12,19,20,17)(15,23,24,21)(25,27,32,37)(26,28,36,39)(29,30,33,31)(34,35,38,40);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,31,38,32,39,30,34)(26,33,35,27,36,29,40,37);(1,4,6,5)(2,7,3,8)(12,20)(15,24)(17,19)(21,23)(25,29)(26,34)(27,30)(28,35)(31,37)(32,33)(36,38)(39,40);(1,5,6,4)(2,7,3,8)(11,18)(14,22)(17,19)(21,23)(25,30,32,31)(26,35,36,40)(27,29,37,33)(28,34,39,38)

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=g,d^2=f,e^2=fi,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^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7178889);