#7200360

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#36091	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2468 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7107556);