#128319

order	512
nilpotency class	4
isoclinism	isoclinic class of #2121
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂³	C₂×C₄×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	#1459
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

544 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ci,f^a=fg,d^b=dg,g^b=gi,f^d=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35515);