#94189

order	512
nilpotency class	5
isoclinism	isoclinic class of #94188
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1390
commutator	C₂×C₄⋊C₄	C₄²
Frattini, cosocle	#2297	C₂²
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2297 ◁ G
LECS	G ▷ #2297 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 344 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fg,e^2=i,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efgi,f^a=fh,g^a=gh,h^a=hi,c^b=ci,d^b=dg,f^b=fi,g^b=gi,h^b=hi,d^c=dhi,f^c=fi,g^c=gi,f^d=fi,g^d=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, 1385);