#479869

order	512
nilpotency class	3
isoclinism	isoclinic class of #478195
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2842
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 596 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=f,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,f^b=fi,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,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, 387065);