#149793

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	#11507
prime radical	#3444	C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₄² ◁ C₂×C₄² ◁ #2920 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 352 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:

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

Polycyclic presentation:

2^9|a^2=e,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,d^b=dhi,e^b=efh,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, 56989);