#133079

order	512
nilpotency class	4
isoclinism	isoclinic class of #133077
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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×#2631	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1075 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 1075 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)(3,13,15,8)(4,17,19,7)(5,10)(6,12)(9,18,22,32)(11,20)(14,26,28,21)(16,24,30,25)(23,27,31,29);(2,7)(4,18)(8,21)(9,10)(11,22)(12,17)(13,23)(14,27)(19,32)(24,26)(25,31)(28,29);(1,3,6,16)(2,8,12,25)(4,14,19,29)(5,15,20,30)(7,21,17,31)(9,23,22,26)(10,13,11,24)(18,27,32,28)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,d^b=dh,f^b=fgh,f^c=fh,f^d=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, 40275);