#477219

order	512
nilpotency class	3
isoclinism	isoclinic class of #476175
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

808 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 808 conjugacy classes of subgroups — too many to draw. Only its 170 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)(4,7)(9,10,13,18,14,19,28,33)(11,15,21,30,22,17,27,34)(12,23,24,32,25,35,26,16)(20,36,37,31,38,40,39,29);(1,2,5,6)(3,4,8,7)(9,11,13,21,14,22,28,27)(10,15,18,30,19,17,33,34)(12,20,24,37,25,38,26,39)(16,29,23,36,32,31,35,40);(1,3,5,8)(2,7,6,4)(9,12,14,25)(10,16,19,32)(11,20,22,38)(13,26,28,24)(15,29,17,31)(18,35,33,23)(21,39,27,37)(30,40,34,36);(1,4,5,7)(2,3,6,8)(9,11)(10,17)(12,20)(13,27)(14,22)(15,19)(16,31)(18,30)(21,28)(23,36)(24,39)(25,38)(26,37)(29,32)(33,34)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 384415);