#254308

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	#18384	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

888 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)(3,11)(4,7)(5,10)(6,9)(8,14)(12,15)(13,16)(17,18)(19,26)(20,31)(21,24)(22,25)(23,29)(27,30)(28,32);(2,7)(3,12)(5,6)(8,11)(13,14)(15,16)(17,18)(19,26,28,32)(20,31)(21,24)(22,25)(23,30,27,29);(1,3)(2,8)(4,13)(5,12)(6,14)(7,15)(9,16)(10,11)(17,19)(18,23)(20,30)(21,29)(22,28)(24,32)(25,27)(26,31);(2,7)(3,13)(9,10)(12,14)(17,18)(19,27)(20,31)(21,24)(22,25)(23,28)(26,29)(30,32)

Polycyclic presentation:

2^9|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, 161504);