#479830

order	512
nilpotency class	3
isoclinism	isoclinic class of #478248
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2840
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

935 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 935 conjugacy classes of subgroups — too many to draw. Only its 154 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)(6,8)(9,10)(11,15,21,29)(12,22,24,16)(13,17,26,33)(14,18)(19,34)(20,28)(23,30)(25,31)(27,36)(32,38)(35,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15,18,29)(12,19,24,36)(13,25,26,20)(14,21)(16,27)(17,31)(22,34)(23,37)(28,33)(30,40,32,39)(35,38);(1,3,4,7)(2,6,5,8)(9,12,14,24)(10,16,18,22)(11,19)(13,23,26,38)(15,27)(17,32,33,30)(20,37)(21,36)(25,35)(28,39)(29,34)(31,40);(2,5)(3,7)(9,13)(10,17,18,33)(11,20)(12,23)(14,26)(15,28,29,31)(16,30,22,32)(19,35)(21,25)(24,38)(27,39,34,40)(36,37)

Polycyclic presentation:

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

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, 387026);