#242996

order	512
nilpotency class	3
isoclinism	isoclinic class of #242992
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

615 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)(3,6)(4,8)(5,7)(9,13)(10,15)(11,14)(12,16)(17,18,20,26,22,25,21,23)(19,27,28,31,30,32,29,24);(1,3,5,11)(2,6,7,14)(4,9,12,10)(8,13,16,15)(18,23)(19,28)(20,21)(25,26)(27,32)(29,30);(3,9)(4,12)(6,13)(8,16)(10,11)(14,15)(17,19,20,29,22,30,21,28)(18,24,26,32,25,31,23,27);(2,7)(6,14)(8,16)(13,15)(18,25)(23,26)(24,31)(27,32)

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,g^a=gh,c^b=cg,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, 150192);