#391743

order	512
nilpotency class	3
isoclinism	isoclinic class of #213744
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1040 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1040 conjugacy classes of subgroups — too many to draw. Only its 234 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=i,d^2=i,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=def,f^b=fi,g^b=gi,d^c=de,h^c=hi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298939);