#94106

order	512
nilpotency class	5
isoclinism	isoclinic class of #1406
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#353
commutator	C₄²	C₄×C₈
Frattini, cosocle	#2285	C₂²
socle	C₂²	#1393
prime radical	C₂⁴	#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2285 ◁ G
LECS	G ▷ #2285 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

298 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 40 points:

(1,2,3,5,4,6,7,8)(9,10,12,17)(11,20,22,36)(13,16,27,32)(14,19,25,34)(15,24,29,23)(18,26,35,28)(21,31,37,33)(30,39,40,38);(9,11)(10,15)(12,24,25,39)(13,23)(14,21)(16,31)(17,33,34,36)(18,20)(19,30)(22,27,37,26)(28,38)(29,32,40,35)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=h,d^2=g,f^2=i,b^a=bc,c^a=cf,e^a=ehi,f^a=fh,h^a=hi,c^b=ci,d^b=dfhi,d^c=dhi,e^d=ei,f^d=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, 1302);