#7271401

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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1344 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)(27,33)(28,30)(29,34)(31,38)(32,35)(36,40)(37,39);(1,2)(3,8)(4,6)(5,7)(9,11,13,18)(10,14,16,22)(12,19,20,17)(15,23,24,21)(25,27,32,36)(26,33,35,40)(28,37,38,34)(29,30,39,31);(9,12)(10,15)(11,17)(13,20)(14,21)(16,24)(18,19)(22,23)(25,28,30,26,32,38,31,35)(27,34,39,40,36,37,29,33);(1,3,5,4)(2,6,7,8)(12,20)(15,24)(17,19)(21,23)(25,29)(26,34)(27,30)(28,33)(31,36)(32,39)(35,37)(38,40);(1,4,5,3)(2,6,7,8)(11,18)(14,22)(17,19)(21,23)(25,30,32,31)(26,28,35,38)(27,29,36,39)(33,34,40,37)

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=gi,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=eh,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, 7178597);