#138298

order	512
nilpotency class	4
isoclinism	isoclinic class of #125300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1505
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 420 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) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45494);