#142996

order	512
nilpotency class	4
isoclinism	isoclinic class of #142966
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	C₂²×C₄⋊C₄	C₂³
prime radical	#3492	C₂²
LCS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

537 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 537 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 64

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

(1,2,6,4)(3,9,12,16)(5,8,14,13)(7,15,10,11)(18,22)(19,26)(23,32)(24,27)(25,31)(28,30);(1,3)(2,7)(4,10)(5,11)(6,12)(8,9)(13,16)(14,15)(17,18)(19,27)(20,23)(21,24)(22,26)(25,29)(28,32)(30,31);(1,4,6,2)(3,10,12,7)(5,8,14,13)(9,11,16,15)(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,f^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,f^b=fg,h^b=hi,d^c=dh,f^c=fg,f^d=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50192);