#101228

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

510 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 94 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=gh,f^2=ghi,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, 8424);