#94474

order	512
nilpotency class	5
isoclinism	isoclinic class of #94438
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1393
commutator	C₂×C₄²	C₄²
Frattini, cosocle	C₄×#124	C₂²
prime radical	C₂²×D₄	C₄⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ C₄×#124 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

222 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 44 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=h,d^2=fi,e^2=i,g^2=i,b^a=bc,c^a=cf,e^a=egh,c^b=cg,d^b=dfh,g^b=gh,h^b=hi,e^c=ehi,e^d=ei,g^e=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, 1670);