#133138

order	512
nilpotency class	4
isoclinism	isoclinic class of #133096
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×#2819	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 993 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)(3,14,16,28)(4,18,17,9)(5,20,23,10)(6,13)(7,12)(8,15,26,21)(11,22)(19,25,31,27)(24,29,32,30);(1,3)(2,8,12,27)(4,7)(5,21)(6,17)(9,20,28,10)(11,25,13,26)(14,29,18,30)(15,24)(16,22)(19,32)(23,31);(1,4,7,3)(2,9,12,28)(5,19,23,15)(6,16,22,17)(8,10,27,20)(11,18,13,14)(21,24,31,32)(25,29,26,30)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,c^b=ci,d^b=dh,f^b=fghi,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, 40334);