#6555584

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533195
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	#24441	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1583 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6462780);