#417311

order	512
nilpotency class	3
isoclinism	isoclinic class of #416472
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2864
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15249
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

597 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 597 conjugacy classes of subgroups — too many to draw. Only its 168 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:

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

Polycyclic presentation:

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

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, 324507);