#254320

order	512
nilpotency class	3
isoclinism	isoclinic class of #254299
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	#3505	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

867 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ehi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=ghi,d^c=di,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, 161516);