#391736

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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1212 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1212 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=def,f^b=fi,d^c=de,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, 298932);