#495316

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

522 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,20,16,21,15,37)(11,17,25,41,23,39,26,55)(12,27,31,58,33,59,32,63)(13,34,35,46,36,64,29,19)(18,42,44,62,45,61,57,30)(22,47,50,71,28,56,51,72)(24,53,54,67,52,69,48,40)(38,65,66,60,43,68,70,49);(2,5)(6,8)(9,11,14,25,16,23,15,26)(10,17,20,41,21,39,37,55)(12,28,31,51,33,22,32,50)(13,24,35,54,36,52,29,48)(18,43,44,70,45,38,57,66)(19,40,34,53,46,67,64,69)(27,56,58,72,59,47,63,71)(30,60,42,68,62,49,61,65);(9,12,29,61,15,32,36,62,16,33,35,42,14,31,13,30)(10,18,34,58,20,44,46,59,21,45,64,63,37,57,19,27)(11,22,48,68,26,51,52,60,23,28,54,65,25,50,24,49)(17,38,53,71,41,66,67,56,39,43,69,72,55,70,40,47);(1,3,4,7)(2,6,5,8)(11,23)(17,39)(22,28)(24,52)(25,26)(38,43)(40,67)(41,55)(47,56)(48,54)(49,60)(50,51)(53,69)(65,68)(66,70)(71,72)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402512);