#504171

order	512
nilpotency class	4
isoclinism	isoclinic class of #501868
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1771 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=f,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,h^a=hi,c^b=cg,d^b=dh,e^b=eh,h^b=hi,e^d=ehi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322593
maximal normal subgroup of	#72322593

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411367);