#238910

order	512
nilpotency class	3
isoclinism	isoclinic class of #238905
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#394
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

860 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 146106);