#254457

order	512
nilpotency class	3
isoclinism	isoclinic class of #254419
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#120	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

843 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:

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

Polycyclic presentation:

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

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, 161653);