#150291

order	512
nilpotency class	5
isoclinism	isoclinic class of #150183
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#353
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 ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂×#391 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 415 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:

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

Polycyclic presentation:

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

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, 57487);