#101518

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	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

405 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 405 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=g,d^2=gh,e^2=h,f^2=ghi,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, 8714);