#146783

order	512
nilpotency class	4
isoclinism	isoclinic class of #12826
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1804
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1050 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)(5,6,10,16,12,17,11,26)(7,18,21,29,22,34,27,30)(8,15,9,25,24,35,23,14)(13,28,31,36,32,20,33,19);(2,4)(5,7)(6,13,17,32)(8,20)(9,19)(10,21)(11,27)(12,22)(14,29,25,30)(15,34,35,18)(16,31,26,33)(23,36)(24,28);(6,14)(8,23)(9,24)(10,11)(13,29)(15,26)(16,35)(17,25)(18,31)(19,28)(20,36)(21,27)(30,32)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231475, #72231505, #72231520, …
maximal normal subgroup of	#72231475, #72231505, #72231520, …
maximal quotient of	#72264879, #72279574, C₂×#146783

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53979);