#101497

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	C₄³	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 420 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) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gh,e^2=h,f^2=hi,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, 8693);