#137201

order	512
nilpotency class	4
isoclinism	isoclinic class of #12173
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

546 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44397);