#7271421

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	#42518	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1386 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,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, 7178617);