#233838

order	512
nilpotency class	3
isoclinism	isoclinic class of #233836
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60798	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1412 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=ghi,e^c=ei,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141034);