#238292

order	512
nilpotency class	3
isoclinism	isoclinic class of #238291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

830 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)(3,11,14,16)(4,8)(5,9)(6,10)(7,13,12,15)(17,18)(19,22)(20,28)(21,25)(23,27)(24,30)(26,31)(29,32);(1,2)(3,12,14,7)(4,8)(5,9)(6,10)(11,13,16,15)(17,19)(18,22)(20,27)(21,26)(23,28)(24,32)(25,31)(29,30);(1,3)(2,7)(4,13)(5,15)(6,14)(8,11)(9,16)(10,12)(17,19)(18,23)(20,29)(21,26)(22,28)(24,31)(25,32)(27,30);(1,4,6,5)(2,8,10,9)(3,13,14,15)(7,11,12,16)(19,26)(22,31)(23,32)(27,29)

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=h,f^2=hi,b^a=be,c^a=cf,f^a=fi,g^a=gh,c^b=cg,d^b=di,g^b=gh,d^c=di,e^c=eh,f^c=fhi

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, 145488);