#238279

order	512
nilpotency class	3
isoclinism	isoclinic class of #238277
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×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

968 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)(3,6)(4,11,13,7)(5,8)(9,16,12,14)(10,15)(17,18,20,22)(19,25,26,32)(21,29,28,24)(23,30,31,27);(1,3,5,10)(2,6,8,15)(4,12)(7,14)(9,13)(11,16)(18,22)(23,31)(24,29)(25,32);(1,4)(2,7)(3,9)(5,13)(6,14)(8,11)(10,12)(15,16)(17,19)(18,23)(20,26)(21,30)(22,31)(24,32)(25,29)(27,28);(3,10)(4,13)(6,15)(7,11)(19,26)(23,31)(25,32)(27,30)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145475);