#508776

order	512
nilpotency class	4
isoclinism	isoclinic class of #508667
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

560 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=efh,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,e^b=ei,g^b=gh,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gh,h^c=hi,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, 415972);