#128353

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	C₂⁴	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 414 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ci,d^a=di,f^a=fg,d^b=dg,e^b=ei,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, 35549);