#253287

order	512
nilpotency class	3
isoclinism	isoclinic class of #253267
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	#2617	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

850 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:

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

Polycyclic presentation:

2^9|b^2=ehi,c^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gh,c^b=cg,d^b=di,g^b=gh,e^c=eh,f^c=fi,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, 160483);