#149869

order	512
nilpotency class	5
isoclinism	isoclinic class of #149793
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2126
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#423	C₂³
socle	C₂	#11532
prime radical	C₂³	#413
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2960 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 229 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57065);