#508802

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	#3150	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 507 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,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, 415998);