#234221

order	512
nilpotency class	3
isoclinism	isoclinic class of #234214
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

1359 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:

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

Polycyclic presentation:

2^9|a^2=h,b^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,d^b=dh,e^b=eh,f^b=fi,d^c=dh,e^c=eh,f^c=fi

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, 141417);