#487959

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	#43361	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

985 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 985 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=gi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395155);