#504174

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×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1771 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

546 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)(7,20,9,29)(8,24)(10,19)(11,18)(12,17)(13,22,15,30)(14,26)(16,28)(21,36,25,33)(23,34,31,32)(27,35);(1,2)(3,4)(5,7,10,23,11,9,12,31)(6,13,17,33,18,15,19,36)(8,25,26,30,27,21,28,22)(14,34,24,20,16,32,35,29);(2,4)(5,8,12,28,11,27,10,26)(6,14,19,35,18,16,17,24)(7,21,31,30,9,25,23,22)(13,32,36,20,15,34,33,29);(5,9,11,7)(6,15,18,13)(8,25,27,21)(10,31,12,23)(14,34,16,32)(17,36,19,33)(20,35,29,24)(22,26,30,28)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=fi,d^2=i,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	#72250290, #72252775, #72252788, …
maximal normal subgroup of	#72250290, #72252775, #72252788, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411370);