#7569667

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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

1868 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=fg,d^2=fgi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257431, #72257516, #72264023
maximal normal subgroup of	#72257431, #72257516, #72264023
maximal quotient of	#72308369

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7476863);