#142405

order	512
nilpotency class	4
isoclinism	isoclinic class of #142404
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1616
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

386 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2)(3,4)(5,6,12,17)(7,19,22,28)(8,15,24,31)(9,14,25,32)(10,18,11,16)(13,23,30,20)(21,33,27,35)(26,29,36,34);(5,7,11,27,12,22,10,21)(6,13,18,34,17,30,16,29)(8,23,25,36,24,20,9,26)(14,28,31,35,32,19,15,33);(2,4)(5,8,11,25,12,24,10,9)(6,14,18,31,17,32,16,15)(7,20,21,36,22,23,27,26)(13,28,29,33,30,19,34,35)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dgi,g^b=gi,h^b=hi,d^c=dh,f^c=fi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250333, #72250350, #72250353
maximal normal subgroup of	#72250333, #72250350, #72250353
maximal quotient of	#72265332

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49601);