#512309

order	512
nilpotency class	4
isoclinism	isoclinic class of #511896
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

603 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 603 conjugacy classes of subgroups — too many to draw. Only its 156 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 — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,d^2=eh,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gh,h^a=hi,c^b=ci,g^b=gi,g^c=gi,e^d=ei,f^d=fi,g^d=gh,h^d=hi

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, 419505);