#101202

order	512
nilpotency class	3
isoclinism	isoclinic class of #100939
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

498 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=gh,d^2=gh,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,f^c=fi

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, 8398);