#392081

order	512
nilpotency class	3
isoclinism	isoclinic class of #391902
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14676
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

631 conjugacy classes of subgroups; only the 232 normal subgroups are shown.

normal subgroups only

This group has 631 conjugacy classes of subgroups — too many to draw. Only its 232 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299277);