#128401

order	512
nilpotency class	4
isoclinism	isoclinic class of #11522
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	#1464
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

406 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 114 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)(7,15)(8,13)(9,14)(10,12)(11,17)(16,23)(18,24)(19,28)(20,25)(21,27)(22,26)(29,32)(30,36)(31,35)(33,34);(5,7,8,18,9,19,21,33,10,17,20,32,22,16,30,31)(6,11,13,24,14,23,27,34,12,15,25,29,26,28,36,35);(1,3,4,2)(6,12)(7,16,17,19)(8,20)(11,23,15,28)(14,26)(18,33,32,31)(21,30)(24,35,29,34)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35597);