#101622

order	512
nilpotency class	3
isoclinism	isoclinic class of #101519
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

485 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 485 conjugacy classes of subgroups — too many to draw. Only its 88 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)(3,5,7,8)(4,6)(10,14)(11,18)(12,15)(17,22)(19,20)(21,23)(25,26,29,31,30,34,28,33)(27,35,37,32,38,39,36,40);(1,3,4,7)(2,5)(6,8)(9,10,15,22,13,17,12,14)(11,16,18,23,20,24,19,21)(26,31)(28,29)(32,35)(33,34)(36,37)(39,40);(9,11,13,20)(10,16,17,24)(12,19,15,18)(14,21,22,23)(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=dfg,c^2=gh,d^2=gh,e^2=gh,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=egh,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, 8818);