#417431

order	512
nilpotency class	3
isoclinism	isoclinic class of #416736
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2862
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15256
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

673 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 673 conjugacy classes of subgroups — too many to draw. Only its 192 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=fg,d^2=gh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324627);