#234436

order	512
nilpotency class	3
isoclinism	isoclinic class of #234422
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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1301 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:

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

Polycyclic presentation:

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

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, 141632);