#93620

order	512
nilpotency class	4
isoclinism	isoclinic class of #93600
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1307
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₄×#97	C₂²
socle	C₂	#7043
prime radical	C₂³	H(3,ℤ/4)
LCS	G ▷ C₂²×C₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₄×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

165 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 165 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=f,e^2=fg,h^2=i,b^a=bc,c^a=cf,e^a=efghi,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,h^b=hi,e^c=ei

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, 816);