#133229

order	512
nilpotency class	4
isoclinism	isoclinic class of #133093
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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

862 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 862 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,g^a=gi,d^b=dh,f^b=fgh,f^c=fhi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293037
maximal normal subgroup of	#72293037
maximal quotient of	#72293163

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40425);