#104232

order	512
nilpotency class	3
isoclinism	isoclinic class of #104181
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

715 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fghi,c^b=cf,d^b=dg,e^b=egi,d^c=dh,e^c=ei

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11428);