#415754

order	512
nilpotency class	3
isoclinism	isoclinic class of #397902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

851 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 851 conjugacy classes of subgroups — too many to draw. Only its 242 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=gh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,f^b=fi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322950);