#148107

order	512
nilpotency class	4
isoclinism	isoclinic class of #136735
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1879
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

706 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,f^b=fhi,h^b=hi,e^c=ehi,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280589, #72280679
maximal normal subgroup of	#72280589, #72280679

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55303);