#144416

order	512
nilpotency class	4
isoclinism	isoclinic class of #144369
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂²×#41	C₂³
prime radical	#23904	C₂
LCS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

796 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fghi,c^b=cf,e^b=eh,g^b=gi,d^c=dg,e^c=ei,f^d=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51612);