#134490

order	512
nilpotency class	4
isoclinism	isoclinic class of #124851
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1505
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 419 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)(7,18,21,31)(8,14)(9,24,27,13)(10,16)(11,17)(12,19,30,22)(15,26,32,23)(20,29)(25,33)(28,36)(34,35);(1,2,3,4)(5,7)(6,12,17,30)(8,19)(9,25)(10,22)(11,21)(13,28,24,34)(14,18,16,31)(15,33,32,29)(20,27)(23,36)(26,35);(6,13)(8,10)(9,26)(12,28)(14,15)(16,32)(17,24)(18,33)(19,22)(20,36)(23,27)(25,35)(29,31)(30,34)

Polycyclic presentation:

2^9|a^2=f,b^2=dh,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eh,f^c=fghi,g^c=gi,h^c=hi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307251, #72307318, #72307331
maximal normal subgroup of	#72307251, #72307318, #72307331

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41686);