#254464

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	#24142	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

899 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)(18,23)(20,21)(24,26)(25,32)(27,31)(28,30);(2,7)(3,12)(5,6)(8,11)(13,14)(15,16)(18,24,26,23)(25,32,27,31);(1,3)(2,8)(4,13)(5,12)(6,14)(7,15)(9,16)(10,11)(17,18)(19,27)(20,24)(21,23)(22,26)(25,29)(28,32)(30,31);(2,7)(3,13)(9,10)(12,14)(17,19)(18,25)(20,30)(21,28)(22,29)(23,31)(24,32)(26,27)

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,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, 161660);