#495838

order	512
nilpotency class	4
isoclinism	isoclinic class of #483873
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

678 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286580, #72286679, D₄×#2180, …
maximal normal subgroup of	#72286580, #72286679, D₄×#2180, …
maximal quotient of	#72287188, #72287440, #72309508, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403034);