#150225

order	512
nilpotency class	5
isoclinism	isoclinic class of #150220
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	#29725	C₂
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ #2311 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 408 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,h^a=hi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,d^c=di,e^d=ehi,f^e=fi

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, 57421);