#142516

order	512
nilpotency class	4
isoclinism	isoclinic class of #142484
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	#29679	C₂
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

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

normal subgroups only

This group has 499 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=gi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,f^a=fhi,c^b=cf,d^b=dg,f^b=fi,h^b=hi,d^c=dh,e^c=egi,f^c=fi,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, 49712);