#487661

order	512
nilpotency class	4
isoclinism	isoclinic class of #487659
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
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1071 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)(10,16)(12,23)(13,28)(14,20)(15,25)(17,22)(18,21)(19,26)(24,30)(27,31)(29,32);(2,7)(3,12)(4,16)(8,22)(9,10)(11,21)(13,15)(14,29)(17,30)(18,31)(20,23)(24,28)(25,32)(26,27);(1,3,6,12)(2,8,11,23)(4,17,18,14)(5,15,19,13)(7,20,21,22)(9,25,26,28)(10,24,27,32)(16,29,31,30);(2,9)(3,13)(7,10)(8,23)(11,26)(12,15)(14,29)(17,30)(20,22)(21,27)(24,32)(25,28)

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,c^b=ci,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283435
maximal normal subgroup of	#72283435
maximal quotient of	#72284362, #72284450

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394857);