#241405

order	512
nilpotency class	3
isoclinism	isoclinic class of #234079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

814 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148601);