#450837

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

754 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 754 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=efh,b^2=f,c^2=e,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi,h^d=hi

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, 358033);