#101342

order	512
nilpotency class	3
isoclinism	isoclinic class of #101243
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

409 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 409 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=gh,d^2=gh,e^2=g,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eg,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, 8538);