#142253

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
prime radical	#29748	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#408 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 723 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)(3,5)(4,6)(7,8)(10,16)(11,21)(12,26)(13,14)(17,33)(18,27)(19,31)(20,32)(22,37)(23,39)(24,25)(28,29)(30,34)(35,38);(5,8)(9,10,15,20)(11,22,25,34)(12,27,29,31)(13,19,14,18)(16,28,32,26)(17,21,35,24)(23,30,39,37)(33,36,38,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,17)(12,23)(13,24)(14,21)(15,25)(16,30)(18,22)(19,34)(20,35)(26,36)(27,33)(28,40)(29,39)(31,38)(32,37)

Polycyclic presentation:

2^9|b^2=i,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, 49449);