#122846

order	512
nilpotency class	4
isoclinism	isoclinic class of #122724
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	#28551	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 366 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) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=gh,d^2=gi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,c^b=cfhi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=dhi,e^d=ehi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30042);