#234207

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	#35293	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1432 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:

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

Polycyclic presentation:

2^9|a^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,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, 141403);