#487621

order	512
nilpotency class	4
isoclinism	isoclinic class of #487619
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	#18162	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

1030 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)(3,8)(4,7)(5,9)(6,11)(10,15)(12,22)(13,19)(14,23)(16,30)(17,20)(18,24)(21,26)(25,29)(27,32)(28,31);(2,7)(4,15)(8,19)(9,10)(11,20)(13,26)(16,28)(17,29)(21,22)(23,30)(24,25)(31,32);(1,3,6,14)(2,8,11,23)(4,16,17,13)(5,12,18,27)(7,19,20,30)(9,22,24,32)(10,21,25,31)(15,28,29,26);(2,9)(3,12)(7,10)(11,24)(13,26)(14,27)(16,28)(20,25)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283025
maximal normal subgroup of	#72283025
maximal quotient of	#72284362

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394817);