#290069

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	#2541	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

(1,2,5,8)(3,10,13,16)(4,9,14,7)(6,15,11,12)(17,18,21,26,22,27,20,25)(19,23,28,24,30,32,29,31);(1,2,4,7,5,8,14,9)(3,6,12,10,13,11,15,16)(17,19,20,29,22,30,21,28)(18,23,25,31,27,32,26,24);(1,2)(3,11)(4,7)(5,8)(6,13)(9,14)(10,15)(12,16)(17,19,22,30)(18,24,27,31)(20,28,21,29)(23,25,32,26);(1,3,4,12,5,13,14,15)(2,6,7,10,8,11,9,16)(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=i,d^2=ei,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, 197265);