#94493

order	512
nilpotency class	5
isoclinism	isoclinic class of #94427
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1390
commutator	C₂²×Q₈	C₄²
Frattini, cosocle	#2297	C₂²
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2297 ◁ G
LECS	G ▷ #2297 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 300 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) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1689);