#488173

order	512
nilpotency class	4
isoclinism	isoclinic class of #487723
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×C₄⋊C₄	C₂⁴
prime radical	#44242	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 918 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^c=fi,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, 395369);