#149718

order	512
nilpotency class	5
isoclinism	isoclinic class of #13609
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#423	C₂³
socle	C₂	#11479
prime radical	#3150	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2932 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

285 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 285 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

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,c^b=ch,d^b=dhi,e^b=efh,f^b=fi,d^c=di,e^d=ehi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56914);