#7544215

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	#28584	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1595 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(17,18,21,22)(19,23,26,27)(20,24,29,30)(25,31,28,32);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(18,22)(20,24)(23,27)(25,31)(28,32)(29,30);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,19)(18,23)(20,28)(21,26)(22,27)(24,32)(25,29)(30,31);(10,14)(13,15)(17,20,21,29)(18,24,22,30)(19,25,26,28)(23,31,27,32);(2,5)(6,8)(10,14)(13,15)(19,26)(23,27)(25,28)(31,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=fgi,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7451411);