#253302

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	C₂×#538	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

836 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,d^2=hi,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, 160498);