#238378

order	512
nilpotency class	3
isoclinism	isoclinic class of #238377
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

991 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)(3,5)(4,11,13,16)(6,8)(7,12,9,15)(10,14)(17,18,21,24)(19,22,26,31)(20,27)(23,29)(25,30)(28,32);(1,3,5,2)(4,7)(6,10,14,8)(9,13)(11,12)(15,16)(17,19)(18,22)(20,28,29,25)(21,26)(23,32,27,30)(24,31);(1,4)(2,7)(3,9)(5,13)(6,15)(8,16)(10,11)(12,14)(17,20)(18,23)(19,25)(21,29)(22,30)(24,27)(26,28)(31,32);(20,29)(23,27)(25,28)(30,32)

Polycyclic presentation:

2^9|a^2=eh,b^2=e,f^2=hi,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,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, 145574);