#128322

order	512
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	#1459
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 542 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)(5,6,9,15,11,16,12,10)(7,17,19,13,21,27,20,28)(8,14,22,29,24,30,25,23)(18,31,32,26,34,35,33,36);(5,7,10,17,12,20,16,28,11,21,15,27,9,19,6,13)(8,18,23,31,25,33,30,36,24,34,29,35,22,32,14,26);(1,3,4,2)(5,8,9,22,11,24,12,25)(6,14,15,29,16,30,10,23)(7,18,19,32,21,34,20,33)(13,26,27,35,28,36,17,31)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35518);