#149800

order	512
nilpotency class	5
isoclinism	isoclinic class of #149800
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2129
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#423	C₂³
socle	C₂	#11510
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2921 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 294 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,d^2=hi,e^2=g,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=dh,e^b=efh,f^b=fi,d^c=di,e^c=ei,e^d=ehi,f^e=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, 56996);