#6434253

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
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×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1873 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)(11,13)(12,14)(15,16)(17,18,22,26,24,19,23,27)(20,29,32,21,25,31,30,28);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,24,18)(20,28,25,21)(22,26,23,27)(29,32,31,30);(11,15)(13,16)(17,20)(18,21)(19,28)(22,30)(23,32)(24,25)(26,29)(27,31);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,24,28)(18,25,19,20)(22,31,23,29)(26,30,27,32);(17,22,24,23)(18,26,19,27)(20,30,25,32)(21,29,28,31)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341449);