#101457

order	512
nilpotency class	3
isoclinism	isoclinic class of #101351
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 367 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=h,d^2=gh,e^2=h,f^2=i,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, 8653);