#7271672

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	#42975	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1414 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fh,c^2=gi,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=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, 7178868);