#254418

order	512
nilpotency class	3
isoclinism	isoclinic class of #254371
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	#24142	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

880 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,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,e^c=eh,f^c=fi,g^c=gi,f^d=fh

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, 161614);