#150302

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

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	C₂⁴	#99
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ #2310 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 333 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) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=gh,e^2=i,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=dg,e^b=efghi,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, 57498);