#142759

order	512
nilpotency class	4
isoclinism	isoclinic class of #142612
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	#3497	C₂²
LCS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

537 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,d^2=g,e^2=gi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fi,h^b=hi,d^c=dh,e^c=egi,f^c=fi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49955);