#149699

order	512
nilpotency class	5
isoclinism	isoclinic class of #1408
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#353
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#423	C₂³
socle	C₂	#11475
prime radical	#3147	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

340 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=e,c^2=g,d^2=hi,e^2=g,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,d^b=dhi,e^b=efh,f^b=fi,e^d=ehi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56895);