#485853

order	512
nilpotency class	4
isoclinism	isoclinic class of #485611
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	#46507	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

923 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 923 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393049);