#299839

order	512
nilpotency class	3
isoclinism	isoclinic class of #299805
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	#62381	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

873 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207035);