#133118

order	512
nilpotency class	4
isoclinism	isoclinic class of #133080
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×#392	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 892 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)(3,12,14,8)(4,9,18,26)(5,11)(6,10)(7,13,21,30)(15,27,29,23)(16,22,31,20)(17,24)(19,25,28,32);(1,3,5,15)(2,7)(4,16,18,30)(6,14,17,29)(8,9)(10,20)(11,22)(12,25)(13,28,31,19)(21,24)(23,26)(27,32);(1,3,5,15)(2,8,11,23)(4,13,18,31)(6,14,17,29)(7,9,22,26)(10,12,24,27)(16,28,30,19)(20,25,21,32)

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=eh,g^a=gi,d^b=dh,f^b=fghi,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, 40314);