#300530

order	512
nilpotency class	3
isoclinism	isoclinic class of #300501
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	#19014	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

917 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=f,d^2=ehi,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^b=gi,d^c=dhi,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207726);