#103196

order	512
nilpotency class	3
isoclinism	isoclinic class of #103161
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₁

782 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 782 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,c^b=cf,d^b=dg,e^b=ei,d^c=dh

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, 10392);