#253285

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	#18284	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

952 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)(3,10)(4,8)(5,7)(6,11)(9,13)(12,15)(14,16)(17,18)(19,21)(20,27)(22,26)(23,28)(24,30)(25,31)(29,32);(1,3)(2,6)(4,11)(5,12)(7,14)(8,10)(9,15)(13,16)(17,19,21,18)(20,23)(22,26,30,24)(25,29)(27,28)(31,32);(2,7)(5,13)(6,14)(8,9)(10,15)(12,16)(17,20)(18,23)(19,25)(21,29)(22,31)(24,32)(26,27)(28,30);(2,8)(3,11)(7,9)(12,16)(18,19)(22,30)(23,25)(28,31)

Polycyclic presentation:

2^9|b^2=ehi,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, 160481);