#512073

order	512
nilpotency class	4
isoclinism	isoclinic class of #512072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

953 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=eh,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,h^a=hi,d^b=di,f^b=fi,e^c=ei,f^c=fhi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250086, #72250096, #72250114
maximal normal subgroup of	#72250086, #72250096, #72250114

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419269);