#453942

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2804
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

710 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 162 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=efg,c^2=egh,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361138);