#495666

order	512
nilpotency class	3
isoclinism	isoclinic class of #188759
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

447 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 447 conjugacy classes of subgroups — too many to draw. Only its 194 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,d^2=f,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,c^b=ch,e^b=ei,d^c=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402862);