#290074

order	512
nilpotency class	3
isoclinism	isoclinic class of #290011
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

609 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=efh,c^2=hi,d^2=ehi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=di,g^b=gh,d^c=dh,f^c=fi,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, 197270);