#7533313

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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	#45356	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1687 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7440509);