#234192

order	512
nilpotency class	3
isoclinism	isoclinic class of #234178
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	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1329 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 141388);