#430484

order	512
nilpotency class	3
isoclinism	isoclinic class of #417690
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

1163 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 1163 conjugacy classes of subgroups — too many to draw. Only its 182 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=efi,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,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, 337680);