#101356

order	512
nilpotency class	3
isoclinism	isoclinic class of #101351
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

392 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 82 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)(11,14)(12,19,20,15)(13,16)(17,21,23,24)(18,22)(25,26,30,33)(27,31)(28,36)(29,34,38,39)(32,35)(37,40);(1,2,4,6,5,7,3,8)(9,11,13,18)(10,14,16,22)(12,17,20,23)(15,21,19,24)(25,27)(26,31,33,36)(28,30)(29,37)(32,34,40,39)(35,38);(1,3,5,4)(2,6,7,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,23)(22,24)(25,28,30,27)(26,32,33,40)(29,35,38,37)(31,39,36,34)

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=g,d^2=gh,e^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eh,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, 8552);