#234274

order	512
nilpotency class	3
isoclinism	isoclinic class of #234257
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

1314 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,16)(14,15)(17,18,23,27)(19,28,20,31)(21,26,22,25)(24,30,32,29);(4,5)(7,15)(8,13)(11,14)(17,19)(18,24)(20,23)(21,30)(22,29)(25,31)(26,28)(27,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,16)(9,15)(10,13)(17,20,23,19)(18,24,27,32)(21,29,22,30)(25,31,26,28);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,16,15,10)(19,20)(24,32)(28,31)(29,30)

Polycyclic presentation:

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

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, 141470);