#6632473

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632074
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	#37271	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1767 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,23)(19,26,27,32)(20,25,22,29)(24,30,31,28);(1,3)(2,5)(4,7)(6,8)(10,13)(11,15)(18,23)(19,27)(25,29)(28,30);(3,7)(5,8)(9,11,12,15)(10,14,13,16)(17,19,21,27)(18,24,23,31)(20,28,22,30)(25,32,29,26);(3,7)(5,8)(10,13)(14,16)(17,18)(19,24)(20,29)(21,23)(22,25)(26,28)(27,31)(30,32);(2,6)(5,8)(17,20,21,22)(18,25,23,29)(19,28,27,30)(24,32,31,26)

Polycyclic presentation:

2^9|a^2=i,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfhi,g^c=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6539669);