#133130

order	512
nilpotency class	4
isoclinism	isoclinic class of #133077
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂×#392	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 929 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,c^b=ci,d^b=dh,f^b=fghi,f^c=fh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293037
maximal normal subgroup of	#72293037
maximal quotient of	#72293156

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40326);