#500716

order	512
nilpotency class	4
isoclinism	isoclinic class of #499763
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1085 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:

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

Polycyclic presentation:

2^9|b^2=e,d^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=ghi,h^b=hi,d^c=di,f^c=fh,g^c=ghi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319933, C₂×#500716
maximal normal subgroup of	#72319933, C₂×#500716
maximal quotient of	C₂×#500716

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407912);