#142245

order	512
nilpotency class	4
isoclinism	isoclinic class of #142244
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	#23390	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

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

normal subgroups only

This group has 753 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)(9,10,15,21)(11,22,25,38)(12,26,30,37)(13,20,14,19)(16,32,35,23)(17,31,36,24)(18,28,33,29)(27,39,40,34);(5,8)(9,11)(10,16)(12,27)(13,24)(14,31)(15,25)(17,33)(18,36)(19,34)(20,39)(21,35)(22,26)(23,29)(28,32)(30,40)(37,38);(1,3)(2,5)(4,7)(6,8)(9,11)(10,17)(12,23)(13,31)(14,24)(15,25)(16,33)(18,35)(19,38)(20,22)(21,36)(26,39)(27,29)(28,40)(30,32)(34,37)

Polycyclic presentation:

2^9|a^2=i,d^2=gi,e^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,g^b=gi,d^c=dh,e^c=ei

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, 49441);