#486143

order	512
nilpotency class	4
isoclinism	isoclinic class of #486139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#23984	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=i,e^2=g,b^a=be,c^a=cf,d^a=dg,e^a=eg,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,e^c=eh,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286366
maximal normal subgroup of	#72286366

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393339);