#242978

order	512
nilpotency class	3
isoclinism	isoclinic class of #242971
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

638 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 150174);