#133097

order	512
nilpotency class	4
isoclinism	isoclinic class of #133096
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂×#2818	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1058 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 1058 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,c^b=ci,d^b=dh,f^b=fgh,f^c=fhi,f^d=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, 40293);