#148358

order	512
nilpotency class	4
isoclinism	isoclinic class of #135928
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1864
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

674 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250132, #72250135, #72250141, …
maximal normal subgroup of	#72250132, #72250135, #72250141, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55554);