#137970

order	512
nilpotency class	4
isoclinism	isoclinic class of #125110
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

575 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 575 conjugacy classes of subgroups — too many to draw. Only its 96 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)(3,4)(5,6,10,17,12,16,11,18)(7,19,23,29,22,34,24,26)(8,25,27,35,28,14,13,32)(9,21,30,33,20,36,31,15);(5,7,20,25,11,23,31,32,12,22,9,14,10,24,30,35)(6,13,33,19,17,8,21,26,16,27,15,34,18,28,36,29);(2,4)(5,8)(6,14)(7,21)(9,29)(10,13)(11,27)(12,28)(15,23)(16,25)(17,35)(18,32)(19,30)(20,26)(22,36)(24,33)(31,34)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,d^2=gh,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,d^c=di,f^c=fi,f^d=fi

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, 45166);