#93639

order	512
nilpotency class	4
isoclinism	isoclinic class of #93526
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1297
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂²×#97	C₂²
socle	C₂	#7046
prime radical	C₂⁴	#108
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂²×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 210 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=fh,e^2=h,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efghi,g^a=gi,c^b=cg,d^b=dh,f^b=fi,e^c=ei,e^d=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, 835);