#148820

order	512
nilpotency class	4
isoclinism	isoclinic class of #13343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	B(4,𝔽₂)
commutator	C₂²×D₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂²	#1889
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

670 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 670 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(2,3)(5,6,10,14,11,15,9,23)(7,12,17,27,18,28,24,26)(8,19,21,30,22,35,25,36)(13,29,32,34,33,20,31,16);(1,2,4,3)(5,7)(6,12)(8,20)(9,24)(10,17)(11,18)(13,30)(14,27)(15,28)(16,21)(19,31)(22,29)(23,26)(25,34)(32,35)(33,36);(1,3,4,2)(5,8)(6,13)(7,16)(9,25)(10,21)(11,22)(12,19)(14,32)(15,33)(17,29)(18,34)(20,24)(23,31)(26,36)(27,30)(28,35)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,f^2=i,g^2=i,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,f^c=fi,h^c=hi,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56016);