#134290

order	512
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×#117	C₂³
socle	C₂²	#1799
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

522 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gi,e^c=ei,f^c=fgi,f^e=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41486);