#123615

order	512
nilpotency class	3
isoclinism	isoclinic class of #2008
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1438
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2474 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)(5,6,9,15,11,16,10,14)(7,17,19,12,20,28,21,29)(8,22,25,31,26,32,24,13)(18,30,35,36,23,27,34,33);(5,7,10,21,11,20,9,19)(6,12,15,28,16,29,14,17)(8,23,24,35,26,18,25,34)(13,30,22,36,31,27,32,33);(2,4)(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,22)(15,32)(16,31)(17,33)(19,34)(20,23)(21,35)(28,36)(29,30)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237177, #72237183, #72237192, …
maximal normal subgroup of	#72237177, #72237183, #72237192, …
maximal quotient of	#72239066

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30811);