#512127

order	512
nilpotency class	4
isoclinism	isoclinic class of #511796
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#3486	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1026 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 144 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)(11,21)(12,18)(13,26)(14,29)(15,20)(16,22)(17,25)(19,31)(23,39)(24,36)(27,34)(28,35)(30,32)(33,40)(37,38);(2,5)(6,8);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,22)(13,27)(14,30)(15,24)(17,32)(18,36)(19,33)(20,34)(21,26)(23,31)(25,40)(28,38)(29,37)(35,39);(9,12,15,13)(10,17,20,35)(11,22,24,27)(14,28,31,25)(16,32,34,39)(18,19,26,29)(21,37,36,33)(23,40,30,38)

Polycyclic presentation:

2^9|c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,h^a=hi,f^c=fi,g^c=ghi,f^d=fh,g^d=gi,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419323);