#94398

order	512
nilpotency class	5
isoclinism	isoclinic class of #94303
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1393
commutator	C₂×M₁₆	C₄²
Frattini, cosocle	#2929	C₂²
socle	C₂	#7287
prime radical	#525	Q₈
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2929 ◁ G
LECS	G ▷ #2929 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

177 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 177 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,5,11,7,13,28,42,8,15,29,12,6,14,27,43)(3,16,18,47,30,38,54,63,21,49,19,48,20,55,53,64)(4,22,24,56,31,40,60,44,26,45,59,46,25,10,37,39)(9,32,33,51,34,58,61,17,36,50,41,23,35,57,62,52);(1,3,5,19,7,30,28,53,8,21,29,18,6,20,27,54)(2,9,12,33,13,34,43,61,15,36,11,41,14,35,42,62)(4,23,24,58,31,52,60,50,26,51,59,57,25,17,37,32)(10,38,39,64,22,49,56,47,40,55,44,63,45,16,46,48)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=fh,e^2=fh,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=di,g^b=gh,e^c=ehi,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1594);