#7194812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1185 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)(3,8)(4,6)(5,7)(9,10,13,17)(11,15,18,14)(12,20,22,16)(19,23,21,24)(26,30)(27,34)(28,37)(31,39)(32,33)(35,38);(9,11,13,18)(10,14,17,15)(12,21,22,19)(16,24,20,23)(25,26)(27,35)(28,32)(29,33)(30,37)(31,34)(36,38)(39,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,24)(20,23)(21,22)(25,27)(26,31)(28,36)(29,34)(30,38)(32,39)(33,35)(37,40);(1,4,5,3)(2,6,7,8)(11,18)(14,15)(19,21)(23,24)(26,32)(27,34)(28,37)(30,33)(31,38)(35,39);(9,12,13,22)(10,16,17,20)(11,19,18,21)(14,23,15,24)(26,33)(27,34)(30,32)(36,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7102008);