#150284

order	512
nilpotency class	5
isoclinism	isoclinic class of #7402
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#353
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#390	C₂³
socle	C₂²	C₂×#353
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2297 ◁ G
LECS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 425 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)(11,20,23,15)(12,16)(13,18,21,17)(14,19)(22,33,36,27)(24,28,34,29)(25,31,35,30)(26,32)(37,38,40,39);(1,2)(3,7)(4,6)(5,8)(9,11,21,34,14,24,13,23)(10,15,18,28,19,29,17,20)(12,22,35,40,26,37,25,36)(16,27,31,38,32,39,30,33);(9,12,14,26)(10,16,19,32)(11,22,24,37)(13,25,21,35)(15,27,29,39)(17,30,18,31)(20,33,28,38)(23,36,34,40)

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=i,d^2=ghi,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,d^b=dg,e^b=efgh,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, 57480);