#150211

order	512
nilpotency class	5
isoclinism	isoclinic class of #150210
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#354
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#390	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2297 ◁ G
LECS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

404 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 74 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)(11,20,23,31)(12,16)(13,18,28,17)(14,19)(15,22,30,27)(21,36,37,32)(24,38,39,29)(25,34,40,33)(26,35);(2,5)(3,8)(6,7)(9,11,14,22)(10,15)(12,24,26,21)(13,27,28,23)(16,32)(17,20)(18,30)(19,31)(25,37,40,39)(29,35)(33,38)(34,36);(9,12,14,26)(10,16,19,35)(11,21,22,24)(13,25,28,40)(15,29,31,32)(17,33,18,34)(20,36,30,38)(23,37,27,39)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293124, #72293189, #72293207
maximal normal subgroup of	#72293124, #72293189, #72293207

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57407);