#6435251

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432838
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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1890 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18)(19,27)(20,26)(21,25)(22,24)(23,29)(28,31)(30,32);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,28)(20,25)(21,26)(22,24)(23,30)(27,31)(29,32);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,19)(18,23)(20,29)(21,31)(22,30)(24,28)(25,27)(26,32);(9,11,12,15)(10,14,13,16)(18,24)(23,28)(25,26)(27,32);(17,20,22,21)(18,25,24,26)(19,29,30,31)(23,27,28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6342447);