#6635559

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632088
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1798 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)(10,14)(11,12)(17,18,19,26,22,20,23,24)(21,28,27,31,30,25,29,32);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(17,19,22,23)(18,24,20,26)(21,27,30,29)(25,31,28,32);(3,7)(5,8)(9,11,13,12)(10,15,14,16)(17,20,22,18)(19,26,23,24)(21,28,30,25)(27,32,29,31);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32);(2,6)(5,8)(17,19,22,23)(18,24,20,26)(21,29,30,27)(25,32,28,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,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, 6542755);