#7623320

order	512
nilpotency class	4
isoclinism	isoclinic class of #7623311
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁵
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1654 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)(3,7)(4,6)(5,8)(9,10,13,18)(11,21,23,16)(12,19,15,31)(14,28,27,32)(17,36,37,24)(20,26,39,30)(22,35,25,33)(29,38,40,34);(1,3)(2,5)(4,8)(6,7)(9,11,13,23)(10,16,18,21)(12,25,15,22)(14,24,27,36)(17,28,37,32)(19,35,31,33)(20,34,39,38)(26,40,30,29);(3,8)(5,7)(9,11)(10,17)(12,22)(13,23)(14,29)(15,25)(16,32)(18,37)(19,38)(20,33)(21,28)(24,30)(26,36)(27,40)(31,34)(35,39);(2,6)(5,7)(10,18)(16,21)(17,37)(19,31)(20,39)(28,32)(33,35)(34,38);(9,12,13,15)(10,19,18,31)(11,22,23,25)(14,26,27,30)(16,33,21,35)(17,38,37,34)(20,28,39,32)(24,29,36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gh,h^a=hi,c^b=cf,e^b=ei,g^c=gh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287359, #72320147, #72320496
maximal normal subgroup of	#72287359, #72320147, #72320496

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7530516);