#300227

order	512
nilpotency class	3
isoclinism	isoclinic class of #300205
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62477	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

963 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)(3,8)(4,14)(5,15)(6,10)(7,13)(9,11)(12,16)(17,18)(19,24)(20,29)(21,30)(22,25)(23,26)(27,31)(28,32);(1,3,11,4,6,12,5,13)(2,7,15,16,10,14,9,8)(17,19,23,28)(18,24,26,32)(20,21,27,22)(25,29,30,31);(3,12)(4,13)(7,14)(8,16)(17,20,21,19,23,27,22,28)(18,24,25,29,26,32,30,31);(1,4,11,12,6,13,5,3)(2,8,15,7,10,16,9,14)

Polycyclic presentation:

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

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, 207423);