#142252

order	512
nilpotency class	4
isoclinism	isoclinic class of #142252
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 787 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49448);