#299607

order	512
nilpotency class	3
isoclinism	isoclinic class of #299601
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

903 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)(3,9,11,16)(4,13,10,7)(6,14,15,12)(17,18)(19,25)(20,28)(21,24)(22,27)(23,29)(26,31)(30,32);(1,3,10,14,5,11,4,12)(2,6,13,16,8,15,7,9)(19,26)(20,29)(23,28)(25,31);(3,11)(6,15)(9,16)(12,14)(17,19,20,27,21,26,29,32)(18,22,23,25,24,30,28,31);(1,3,10,14,5,11,4,12)(2,6,13,16,8,15,7,9)

Polycyclic presentation:

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

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, 206803);