#7105608

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

1597 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=fhi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308985, #72309002
maximal normal subgroup of	#72308985, #72309002

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7012804);