#6599691

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573549
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	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1906 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=ehi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282340
maximal normal subgroup of	#72282340

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6506887);