#132368

order	512
nilpotency class	4
isoclinism	isoclinic class of #132197
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#559 ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

674 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=h,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,e^c=ei,f^c=fgi,f^d=fi,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, 39564);