#142246

order	512
nilpotency class	4
isoclinism	isoclinic class of #142246
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

728 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 70 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)(3,5)(4,6)(7,8)(10,16)(11,21)(12,25)(13,14)(17,34)(18,32)(19,26)(20,33)(22,35)(23,27)(24,30)(28,29)(31,36)(37,40);(5,8)(9,10)(11,22)(12,26)(13,18)(14,19)(15,20)(16,25)(17,21)(23,31)(24,36)(27,35)(28,33)(29,32)(30,37)(34,38)(39,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,17)(12,27)(13,30)(14,21)(15,24)(16,31)(18,22)(19,36)(20,37)(23,29)(25,39)(26,40)(28,38)(32,34)(33,35)

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=fh,g^a=gi,c^b=cf,d^b=dg,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, 49442);