#149706

order	512
nilpotency class	5
isoclinism	isoclinic class of #13603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#354
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#382	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#124 ◁ G
LECS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

298 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56902);